the ordered matrix roots

Eigenvalues: Formula, Verified Results, Charts and Interpretation

An eigenvalue is the variance associated with an eigenvector of the analyzed correlation or covariance matrix. In PCA it is component variance; in factor-retention work it is an input to rules such as the Kaiser criterion, scree plot, and parallel analysis. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.

Ordered rootsRetention boundarySimulation or graphReal data

Eigenvalue 13.195831

Eigenvalue 21.817089

Eigenvalue 31.393698

Eigenvalue 40.846560

Verified result

The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

For Eigenvalues, eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

Interpretive limit: An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

What Eigenvalues measures

The exact estimand and the result this method is allowed to support.

Eigenvalues addresses one defined analytical target: An eigenvalue is the variance associated with an eigenvector of the analyzed correlation or covariance matrix. In PCA it is component variance; in factor-retention work it is an input to rules such as the Kaiser criterion, scree plot, and parallel analysis.

Quantity estimated in this analysis

The ordered matrix roots is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089 supplies the first supporting check. Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

For Eigenvalues, the calculation retains full precision until the final display. That matters because the software reports, spreadsheet formulas, chart labels, and narrative must refer to one identical result rather than separately rounded approximations.

Interpretation that is not permitted

An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

For Eigenvalues, this boundary is substantive. A nearby coefficient may share the same data or model, yet it answers a different question. The article therefore names every supporting statistic instead of using broad labels such as “valid,” “good,” or “significant” without the object being evaluated.

Worked conclusion: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

When to use Eigenvalues

Research scope, neighboring methods, and excluded claims.

Research question answered

The defensible question is whether the ordered matrix roots supports the result stated for the declared dataset and analytical specification. It is answered by verify that the nine roots sum to nine for a correlation-matrix PCA, followed by recalculate eigenpairs from the same variable order. The evidence is bounded by Eigenvalue 1 = 3.195831 and its named companion quantities.

For Eigenvalues, changing the case set, expert panel, item block, estimator, factor count, rotation, baseline model, bootstrap design, or criterion definition changes the question. Such a change requires a new result rather than a revision of the wording around the old value.

Nearest methods that answer different questions

Scree Plot: The scree plot visualizes the ordered eigenvalues and uses the elbow as a qualitative retention cue.

Parallel Analysis: Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size.

These distinctions determine which formula, output table, and chart can legitimately appear in a Eigenvalues post.

Scope limit: An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

Real data used for Eigenvalues

Variables, coding, sample or panel size, and the role each input plays.

For Eigenvalues, the factor-oriented analysis uses 649 complete records and nine ordered variables. TravelAccess is defined as 5 − traveltime so larger values represent easier travel. The correlation matrix, eigenvalues, communalities, loading matrices, rotation output, and simulation cutoffs all preserve the same variable order.

For the ordered matrix roots, the data are not merely background. A change in correlation type, standardization, missing-case rule, variable order, or retained dimension count changes the matrix on which Eigenvalue 1 = 3.195831 was obtained.

Variable	Meaning	Mean	SD	Range	Construct
G1	first-period grade	11.3991	2.7453	0–19	Academic Achievement
G2	second-period grade	11.5701	2.9136	0–19	Academic Achievement
G3	final grade	11.9060	3.2307	0–19	Academic Achievement
Medu	mother’s education	2.5146	1.1346	0–4	Educational Advantage
Fedu	father’s education	2.3066	1.0999	0–4	Educational Advantage
TravelAccess	reverse-coded travel accessibility	3.4314	0.7487	1–4	Educational Advantage
goout	frequency of going out	3.1849	1.1758	1–5	Social-Alcohol Exposure
Dalc	workday alcohol use	1.5023	0.9248	1–5	Social-Alcohol Exposure
Walc	weekend alcohol use	2.2804	1.2844	1–5	Social-Alcohol Exposure

Data-to-result trace: Verify that the nine roots sum to nine for a correlation-matrix pca is the first data-integrity check, followed by recalculate eigenpairs from the same variable order. Both checks are performed before the primary coefficient is interpreted.

Eigenvalues assumptions and design requirements

Six conditions checked before the coefficient or decision rule is interpreted.

1. The matrix is symmetric and based on the intended variables

This condition determines whether the input object matches the formula. In the current Eigenvalues analysis, the check is to verify that the nine roots sum to nine for a correlation-matrix PCA while preserving Eigenvalue 1 = 3.195831.

For Eigenvalues, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.

2. Variables are standardized when a correlation matrix is used

This requirement controls whether the numerical estimate has the interpretation claimed. In the current Eigenvalues analysis, the check is to recalculate eigenpairs from the same variable order while preserving Eigenvalue 2 = 1.817089.

3. The eigenvalues are sorted consistently

This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Eigenvalues analysis, the check is to compare the third and fourth roots with simulated roots while preserving Eigenvalue 3 = 1.393698.

4. Missing data and correlation type are fixed

This specification rule keeps the software routes numerically comparable. In the current Eigenvalues analysis, the check is to inspect the scree elbow around the retained boundary while preserving Eigenvalue 4 = 0.846560.

5. PCA roots are not mislabeled common-factor roots

This diagnostic requirement is checked before a benchmark is applied. In the current Eigenvalues analysis, the check is to avoid interpreting eigenvector sign as substantive direction while preserving Three-dimension cumulative variance = 71.1846%.

6. Rounding is not used before retention comparisons

This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Eigenvalues analysis, the check is to separate component variance from common-factor variance while preserving Horn retained factors = 3.

Assumption consequence: An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

Eigenvalues hypotheses or decision rule

The statistical question is stated at the correct level for this method.

Statistical question

The retention decision asks whether each observed ordered root remains larger than its adjacent or simulated reference value; it is not a single omnibus null hypothesis.

Uncertainty is evaluated at the retention boundary, especially the last retained and first rejected dimensions.

Decision for the worked analysis

The calculation yields Eigenvalue 1 = 3.195831. Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Language rule: the conclusion names the tested model, construct pair, item set, retained dimensions, or expert panel. It does not convert nonrejection into proof or a benchmark into a universal pass.

Eigenvalues formula and worked substitution

Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.

The equation below is the defining mathematical object for Eigenvalues. Its symbols are connected to the saved inputs and to Eigenvalue 1 = 3.195831, Eigenvalue 2 = 1.817089, Eigenvalue 3 = 1.393698, Eigenvalue 4 = 0.846560.

Eigenvalues equationsNative MathML · no external script

Eigenvalue equation

$\begin{array}{rcl} R v_{i} & = & λ_{i} v_{i} \\ | R - λ I | & = & 0 \end{array}$

Eigenvalues quantify variance along mutually orthogonal matrix directions.

Derived coefficients, scores, scree points, and Kaiser count

$\begin{array}{rcl} a_{i} & = & \sqrt{\frac{n λ_{i}}{\sum_{j_{1}}^{p}}} \\ x_{i k} & = & \sqrt{λ_{i}} v_{k i} \\ Scree points & = & (i, λ_{i}) \\ k & = & number of λ_{i} > 1 \end{array}$

These expressions are typeset with native fractions, radicals, subscripts, superscripts, and summation limits.

First four observed roots

$\begin{array}{rcl} λ_{1} & = & 3.1958 \\ λ_{2} & = & 1.8171 \\ λ_{3} & = & 1.3937 \\ λ_{4} & = & 0.8466 \end{array}$

Three observed roots exceed one; the fourth falls below one, matching the three-dimension retention result.

Symbol and denominator control

For Eigenvalues, the numerator, denominator, matrix order, degrees of freedom, factor count, or panel size shown in the MathML card is retained exactly. A formula from a neighboring method is not substituted even when both produce values on a similar scale.

Full-precision substitution

The spreadsheet and software outputs retain unrounded inputs until the final displayed value. The arithmetic is then reconciled with Eigenvalue 1 = 3.195831 and Eigenvalue 2 = 1.817089.

Step-by-step Eigenvalues calculation

Every stage is tied to a saved value and a method-specific condition.

The worked calculation follows six operations specific to the ordered matrix roots. Each operation produces a quantity used by the next step, so a discrepancy is resolved where it originates rather than hidden by rounding.

Establish the analytical object

Action: Verify that the nine roots sum to nine for a correlation-matrix PCA.

Numerical trace: Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089.

Condition: the matrix is symmetric and based on the intended variables. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Reconstruct the first required quantity

Action: Recalculate eigenpairs from the same variable order.

Numerical trace: Eigenvalue 2 = 1.817089; Eigenvalue 3 = 1.393698.

Condition: variables are standardized when a correlation matrix is used. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Verify the companion quantity

Action: Compare the third and fourth roots with simulated roots.

Numerical trace: Eigenvalue 3 = 1.393698; Eigenvalue 4 = 0.846560.

Condition: the eigenvalues are sorted consistently. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Apply the decision rule

Action: Inspect the scree elbow around the retained boundary.

Numerical trace: Eigenvalue 4 = 0.846560; Three-dimension cumulative variance = 71.1846%.

Condition: missing data and correlation type are fixed. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Inspect local evidence

Action: Avoid interpreting eigenvector sign as substantive direction.

Numerical trace: Three-dimension cumulative variance = 71.1846%; Horn retained factors = 3.

Condition: PCA roots are not mislabeled common-factor roots. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Reconcile and report

Action: Separate component variance from common-factor variance.

Numerical trace: Horn retained factors = 3; Observed eigenvalue 3 = 1.393698.

Condition: rounding is not used before retention comparisons. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Final reconciliation: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Eigenvalues results and interpretation

Primary and supporting statistics are kept separate and precisely labeled.

Primary result

3.195831

Eigenvalue 1

Why the result is internally coherent

Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

For Eigenvalues, the two quantities are reported together because one is primary and the other supplies context; neither is renamed as the other.

Result item	Exact value	Interpretation restricted to this method
Eigenvalue 1	3.195831	Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Eigenvalue 2	1.817089	Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Eigenvalue 3	1.393698	Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Eigenvalue 4	0.846560	Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Three-dimension cumulative variance	71.1846%	Three-dimension cumulative variance = 71.1846% belongs to the declared matrix and retained dimensions and must not be relabeled as model fit.
Horn retained factors	3	Horn retained factors = 3 is retained as a distinct supporting quantity for the ordered matrix roots; it is not substituted for the primary result.
Observed eigenvalue 3	1.393698	Observed eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Horn 95th percentile root 3	1.106209	Horn 95th percentile root 3 = 1.106209 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Parallel iterations	500	Parallel iterations = 500 documents simulation or convergence effort rather than substantive magnitude.
Observed eigenvalue 4	0.846560	Observed eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Horn 95th percentile root 4	1.064106	Horn 95th percentile root 4 = 1.064106 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.
Overall KMO	0.713439	Overall KMO = 0.713439 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.
Lowest item MSA	0.588169	Lowest item MSA = 0.588169 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.
Highest item MSA	0.867806	Highest item MSA = 0.867806 describes the balance between ordinary and partial correlations, so it informs factorability or item adequacy rather than factor retention.

Maximum defensible claim: An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

Eigenvalues in Python

The Python route calculates or reconstructs the exact named result.

The Python workflow uses the explicit NumPy/Pandas calculation to calculate or extract the ordered matrix roots from the declared data and analytical specification. It must reproduce Eigenvalue 1 = 3.195831 and retain Eigenvalue 2 = 1.817089 as a separate supporting quantity.

The code is read as an executable analysis, not as a printed answer. Its critical verification is to verify that the nine roots sum to nine for a correlation-matrix PCA; the associated design condition is that the matrix is symmetric and based on the intended variables. An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

Python — Eigenvalues

import pandas as pd

import numpy as np

df = pd.read_csv("student-por.csv", sep=";") df["TravelAccess"] = 5 - df["traveltime"] vars9 = ["G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc"] X = df[vars9].dropna() R=X.corr().to_numpy() values,vectors=np.linalg.eigh(R) order=np.argsort(values)[::-1] values=values[order]; vectors=vectors[:,order] loadings=vectors*np.sqrt(values) print("eigenvalues",values) print("explained",values/values.sum()) print("first three loadings",loadings[:,:3])

Python interpretation: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Eigenvalues in R

The R route declares package, estimator, extraction, rotation, or resampling settings.

The R route uses base R and the displayed matrix operations and the displayed arguments to estimate the ordered matrix roots. Package defaults are made explicit because estimator, matrix type, extraction, rotation, baseline, or bootstrap choices can change the result.

R output is reconciled with Eigenvalue 1 = 3.195831 after the analyst recalculate eigenpairs from the same variable order. Agreement is expected only when the case set, variable order, and method settings match the Python and workbook calculations.

R — Eigenvalues

d <- read.csv2("student-por.csv")

d$TravelAccess <- 5 - d$traveltime

vars9 <- c("G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc")

X <- d[vars9]

e <- eigen(cor(X))

print(e$values); print(e$values/sum(e$values))

loadings <- sweep(e$vectors,2,sqrt(e$values),"*")

print(loadings[,1:3])

R interpretation: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Eigenvalues in SPSS or AMOS

The procedure is labeled honestly when base SPSS does not expose the coefficient.

The SPSS or AMOS section shows the procedure that is actually available for the ordered matrix roots. When base SPSS does not expose the coefficient, the syntax prepares the correct matrix or model and the coefficient is obtained through AMOS, MATRIX operations, or a validated integration rather than by renaming a different test.

The output must identify Eigenvalue 1 = 3.195831 and the settings needed to reproduce it. The software review specifically compare the third and fourth roots with simulated roots, while preserving the requirement that the eigenvalues are sorted consistently.

SPSS or AMOS — Eigenvalues

COMPUTE TravelAccess = 5 - traveltime.

EXECUTE.

FACTOR

/VARIABLES G1 G2 G3 Medu Fedu TravelAccess goout Dalc Walc

/MISSING LISTWISE

/PRINT INITIAL KMO EXTRACTION ROTATION

/PLOT EIGEN

/CRITERIA FACTORS(3) ITERATE(500)

/EXTRACTION PAF

/ROTATION OBLIMIN

/METHOD=CORRELATION.

* Read only the Eigenvalues evidence identified in this post.

SPSS or AMOS interpretation: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Eigenvalues in Excel

The workbook exposes source values, intermediate arithmetic, and the final formula.

The Excel workbook is an arithmetic audit for the ordered matrix roots. Named cells retain the inputs, intermediate components, and final formula leading to Eigenvalue 1 = 3.195831; no rounded constant is pasted over a formula cell.

Excel can verify visible calculations and cross-software agreement, but it does not replace estimation, optimization, rotation, or resampling that must occur in statistical software. The workbook therefore focuses on the check to inspect the scree elbow around the retained boundary and documents Eigenvalue 2 = 1.817089 independently.

Excel — Eigenvalues

Data: 649 rows with documented coding.

Inputs: named cells or ranges required only by Eigenvalues.

Calculation: Use the native MathML formula shown above with named ranges for every input

Audit: compare full-precision Excel output with the Python, R, and SPSS/AMOS values.

Decision: reference the exact result and diagnostics; never paste a rounded value over the formula cell.

Excel interpretation: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Eigenvalues charts and visual diagnostics

Each supplied image is interpreted through its own values and analytical purpose.

Every image below is interpreted as part of the same Eigenvalues analysis. The captions identify what the panel contributes, the exact values visible in the result set, and the condition that would invalidate the reading.

Eigenvalues — 01 Eigenvalues Primary Metrics — 01 Eigenvalues Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for **Eigenvalues**. Read **Eigenvalue 1 = 3.195831** beside **Eigenvalue 2 = 1.817089**; the first quantity is not replaced by the second.
The chart is used to verify that the nine roots sum to nine for a correlation-matrix PCA. Its interpretation remains valid only when the matrix is symmetric and based on the intended variables. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

Eigenvalues — 02 Eigenvalues Ordered Eigenvalues — 02 Eigenvalues Ordered Eigenvalues
This panel places the ordered roots around the retention boundary for **Eigenvalues**. Read **Eigenvalue 2 = 1.817089** beside **Eigenvalue 3 = 1.393698**; the first quantity is not replaced by the second.
The chart is used to recalculate eigenpairs from the same variable order. Its interpretation remains valid only when variables are standardized when a correlation matrix is used. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

Eigenvalues — 03 Eigenvalues Eigenvectors — 03 Eigenvalues Eigenvectors
This panel places the ordered roots around the retention boundary for **Eigenvalues**. Read **Eigenvalue 3 = 1.393698** beside **Eigenvalue 4 = 0.846560**; the first quantity is not replaced by the second.
The chart is used to compare the third and fourth roots with simulated roots. Its interpretation remains valid only when the eigenvalues are sorted consistently. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

Eigenvalues — 04 Eigenvalues Correlation Matrix — 04 Eigenvalues Correlation Matrix
This panel shows the cell-level pattern that a single coefficient can conceal for **Eigenvalues**. Read **Eigenvalue 4 = 0.846560** beside **Three-dimension cumulative variance = 71.1846%**; the first quantity is not replaced by the second.
The chart is used to inspect the scree elbow around the retained boundary. Its interpretation remains valid only when missing data and correlation type are fixed. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

Eigenvalues — 05 Eigenvalues Verified Result Summary — 05 Eigenvalues Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for **Eigenvalues**. Read **Three-dimension cumulative variance = 71.1846%** beside **Horn retained factors = 3**; the first quantity is not replaced by the second.
The chart is used to avoid interpreting eigenvector sign as substantive direction. Its interpretation remains valid only when PCA roots are not mislabeled common-factor roots. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

Eigenvalues verification and sensitivity analysis

Six failure modes are checked against the formula, data, output, and charts.

The following diagnostics are not a general checklist. Each one targets a failure mode that can change the calculation or interpretation of Eigenvalues.

1. Verify that the nine roots sum to nine for a correlation-matrix PCA

Begin by verify that the nine roots sum to nine for a correlation-matrix PCA. For the ordered matrix roots, this operation directly connects Eigenvalue 1 = 3.195831 with Eigenvalue 3 = 1.393698. Eigenvalue 1 = 3.195831 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

The governing condition is that the matrix is symmetric and based on the intended variables. If it fails, the primary coefficient may be attached to the wrong input object. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Scree Plot, because The scree plot visualizes the ordered eigenvalues and uses the elbow as a qualitative retention cue.

2. Recalculate eigenpairs from the same variable order

Next, recalculate eigenpairs from the same variable order. For the ordered matrix roots, this operation directly connects Eigenvalue 2 = 1.817089 with Eigenvalue 4 = 0.846560. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

The governing condition is that variables are standardized when a correlation matrix is used. If it fails, the companion statistic may no longer describe the same model or sample. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Parallel Analysis, because Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size.

3. Compare the third and fourth roots with simulated roots

The third verification is to compare the third and fourth roots with simulated roots. For the ordered matrix roots, this operation directly connects Eigenvalue 3 = 1.393698 with Three-dimension cumulative variance = 71.1846%. Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

The governing condition is that the eigenvalues are sorted consistently. If it fails, the decision boundary can move because the required quantity has changed. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Communalities, because Communalities concern variable-level common variance; eigenvalues concern dimension-level variance in the analyzed matrix.

4. Inspect the scree elbow around the retained boundary

After the core arithmetic is stable, inspect the scree elbow around the retained boundary. For the ordered matrix roots, this operation directly connects Eigenvalue 4 = 0.846560 with Horn retained factors = 3. Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

The governing condition is that missing data and correlation type are fixed. If it fails, software agreement can be artificial if unlike definitions are compared. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Scree Plot, because The scree plot visualizes the ordered eigenvalues and uses the elbow as a qualitative retention cue.

5. Avoid interpreting eigenvector sign as substantive direction

A robustness review must avoid interpreting eigenvector sign as substantive direction. For the ordered matrix roots, this operation directly connects Three-dimension cumulative variance = 71.1846% with Observed eigenvalue 3 = 1.393698. Three-dimension cumulative variance = 71.1846% belongs to the declared matrix and retained dimensions and must not be relabeled as model fit.

The governing condition is that PCA roots are not mislabeled common-factor roots. If it fails, a favorable average can conceal a local failure. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Parallel Analysis, because Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size.

6. Separate component variance from common-factor variance

The final reconciliation should separate component variance from common-factor variance. For the ordered matrix roots, this operation directly connects Horn retained factors = 3 with Horn 95th percentile root 3 = 1.106209. Horn retained factors = 3 is retained as a distinct supporting quantity for the ordered matrix roots; it is not substituted for the primary result.

The governing condition is that rounding is not used before retention comparisons. If it fails, the published conclusion can exceed the evidence actually reproduced. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with Communalities, because Communalities concern variable-level common variance; eigenvalues concern dimension-level variance in the analyzed matrix.

#	Verification operation	Condition protected	Saved quantity traced
1	verify that the nine roots sum to nine for a correlation-matrix PCA	the matrix is symmetric and based on the intended variables	Eigenvalue 1 = 3.195831
2	recalculate eigenpairs from the same variable order	variables are standardized when a correlation matrix is used	Eigenvalue 2 = 1.817089
3	compare the third and fourth roots with simulated roots	the eigenvalues are sorted consistently	Eigenvalue 3 = 1.393698
4	inspect the scree elbow around the retained boundary	missing data and correlation type are fixed	Eigenvalue 4 = 0.846560
5	avoid interpreting eigenvector sign as substantive direction	PCA roots are not mislabeled common-factor roots	Three-dimension cumulative variance = 71.1846%
6	separate component variance from common-factor variance	rounding is not used before retention comparisons	Horn retained factors = 3

Diagnostic conclusion: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Failure boundary: An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

Eigenvalues compared with related methods

Differences in estimand, formula, and conclusion determine the correct choice.

Method choice depends on the estimand, model, and data structure. These three comparisons explain why the post uses the Eigenvalues formula and output rather than a nearby procedure.

Scree Plot

The scree plot visualizes the ordered eigenvalues and uses the elbow as a qualitative retention cue.

In the current analysis, Eigenvalue 2 = 1.817089 remains evidence for the ordered matrix roots; it is not relabeled as a Scree Plot result. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

Parallel Analysis

Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size.

In the current analysis, Eigenvalue 3 = 1.393698 remains evidence for the ordered matrix roots; it is not relabeled as a Parallel Analysis result. Eigenvalue 3 = 1.393698 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

Communalities

Communalities concern variable-level common variance; eigenvalues concern dimension-level variance in the analyzed matrix.

In the current analysis, Eigenvalue 4 = 0.846560 remains evidence for the ordered matrix roots; it is not relabeled as a Communalities result. Eigenvalue 4 = 0.846560 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

Selection rule: An eigenvalue is the variance associated with an eigenvector of the analyzed correlation or covariance matrix. In PCA it is component variance; in factor-retention work it is an input to rules such as the Kaiser criterion, scree plot, and parallel analysis.

How to report Eigenvalues

A complete result paragraph includes the value, analytical object, settings, and limitation.

Results paragraph

Eigenvalues was evaluated using the declared data, specification, and software settings. The primary result was Eigenvalue 1 = 3.195831; Eigenvalue 2 = 1.817089 and Eigenvalue 3 = 1.393698 supplied supporting context. The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

The report then states the limitation explicitly: An eigenvalue greater than one is not a universal proof that a factor exists. Eigenvalues depend on the matrix type, scaling, missing-data treatment, and whether total-variance PCA or common-factor analysis is being evaluated.

Settings that must accompany the result

the matrix is symmetric and based on the intended variables; variables are standardized when a correlation matrix is used; the eigenvalues are sorted consistently; missing data and correlation type are fixed.

For Eigenvalues, these details identify the exact version of the analysis and make cross-software reconciliation possible.

Verification actions retained in the record

verify that the nine roots sum to nine for a correlation-matrix PCA; recalculate eigenpairs from the same variable order; compare the third and fourth roots with simulated roots; inspect the scree elbow around the retained boundary.

The final wording is revised only after those operations reproduce the saved values.

Reporting standard: name the statistic, value, analytical object, sample or panel size, method settings, and limitation in the same result paragraph.

16A

Eigenvalues decision scenarios

For Eigenvalues, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.

Boundary-case interpretation: Verify that the nine roots sum to nine for a correlation-matrix PCA

Consider a review in which Eigenvalue 1 = 3.195831 is reproduced but Eigenvalue 2 = 1.817089 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify that the nine roots sum to nine for a correlation-matrix PCA and verify that the matrix is symmetric and based on the intended variables.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Scree Plot only for method selection: The scree plot visualizes the ordered eigenvalues and uses the elbow as a qualitative retention cue. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Input-definition sensitivity: Recalculate eigenpairs from the same variable order

Consider a review in which Eigenvalue 3 = 1.393698 is reproduced but Eigenvalue 4 = 0.846560 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to recalculate eigenpairs from the same variable order and verify that variables are standardized when a correlation matrix is used.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Software-definition reconciliation: Compare the third and fourth roots with simulated roots

Consider a review in which Three-dimension cumulative variance = 71.1846% is reproduced but Horn retained factors = 3 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare the third and fourth roots with simulated roots and verify that the eigenvalues are sorted consistently.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Communalities only for method selection: Communalities concern variable-level common variance; eigenvalues concern dimension-level variance in the analyzed matrix. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Local-chart conflict: Inspect the scree elbow around the retained boundary

Consider a review in which Observed eigenvalue 3 = 1.393698 is reproduced but Horn 95th percentile root 3 = 1.106209 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to inspect the scree elbow around the retained boundary and verify that missing data and correlation type are fixed.

Alternative-method challenge: Avoid interpreting eigenvector sign as substantive direction

Consider a review in which Parallel iterations = 500 is reproduced but Observed eigenvalue 4 = 0.846560 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid interpreting eigenvector sign as substantive direction and verify that PCA roots are not mislabeled common-factor roots.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Replication and reporting decision: Separate component variance from common-factor variance

Consider a review in which Horn 95th percentile root 4 = 1.064106 is reproduced but Overall KMO = 0.713439 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to separate component variance from common-factor variance and verify that rounding is not used before retention comparisons.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Communalities only for method selection: Communalities concern variable-level common variance; eigenvalues concern dimension-level variance in the analyzed matrix. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Boundary-case interpretation: Verify that the nine roots sum to nine for a correlation-matrix PCA

Consider a review in which Lowest item MSA = 0.588169 is reproduced but Highest item MSA = 0.867806 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify that the nine roots sum to nine for a correlation-matrix PCA and verify that the matrix is symmetric and based on the intended variables.

Input-definition sensitivity: Recalculate eigenpairs from the same variable order

Consider a review in which Bartlett chi-square = 3018.238 is reproduced but Eigenvalue 1 = 3.195831 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to recalculate eigenpairs from the same variable order and verify that variables are standardized when a correlation matrix is used.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Parallel Analysis only for method selection: Parallel analysis compares each observed eigenvalue with roots generated from random data of the same size. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Software-definition reconciliation: Compare the third and fourth roots with simulated roots

Consider a review in which Eigenvalue 2 = 1.817089 is reproduced but Eigenvalue 3 = 1.393698 is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare the third and fourth roots with simulated roots and verify that the eigenvalues are sorted consistently.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with Communalities only for method selection: Communalities concern variable-level common variance; eigenvalues concern dimension-level variance in the analyzed matrix. The published conclusion remains The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

Local-chart conflict: Inspect the scree elbow around the retained boundary

Consider a review in which Eigenvalue 4 = 0.846560 is reproduced but Three-dimension cumulative variance = 71.1846% is not. For the ordered matrix roots, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to inspect the scree elbow around the retained boundary and verify that missing data and correlation type are fixed.

Eigenvalues downloads and reproducibility files

All linked files belong to the same analysis and remain on onlineinternetcafe.com.

PYEigenvalues Python reportReproduces Eigenvalue 1 = 3.195831 and documents how to verify that the nine roots sum to nine for a correlation-matrix PCA.Open file →REigenvalues R reportReproduces Eigenvalue 1 = 3.195831 and documents how to recalculate eigenpairs from the same variable order.Open file →SPSSEigenvalues SPSS/AMOS outputReproduces Eigenvalue 1 = 3.195831 and documents how to compare the third and fourth roots with simulated roots.Open file →XLSXEigenvalues Excel workbookReproduces Eigenvalue 1 = 3.195831 and documents how to inspect the scree elbow around the retained boundary.Open file →

The four files belong to one Eigenvalues analysis. Their primary values, variable order, method settings, and chart labels must agree; a mismatch is resolved in the source calculation before the WordPress draft is published.

Eigenvalues frequently asked questions

Answers use the worked result and the exact method boundary.

What does Eigenvalues measure?

What is the main result in this Eigenvalues analysis?

Eigenvalue 1 = 3.195831. The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

What does the result not prove?

Which supporting value should be reported with the primary result?

Eigenvalue 2 = 1.817089 is the first companion quantity. Eigenvalue 2 = 1.817089 is interpreted in rank order and beside adjacent observed or simulated roots before a retention decision is made.

Which assumption is most likely to change the interpretation?

The first requirement is that the matrix is symmetric and based on the intended variables. The result is recomputed if that condition is not satisfied.

What is the most important numerical verification?

The analyst must verify that the nine roots sum to nine for a correlation-matrix PCA. That operation traces Eigenvalue 1 = 3.195831 to the formula and saved inputs.

Why can software packages disagree on Eigenvalues?

Disagreement can arise because variables are standardized when a correlation matrix is used or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.

How is Eigenvalues different from Scree Plot?

The scree plot visualizes the ordered eigenvalues and uses the elbow as a qualitative retention cue.

How should a chart be interpreted?

Each chart is tied to a named output such as Eigenvalue 3 = 1.393698. It supports a local calculation or diagnostic and does not replace the full numerical result.

How should Eigenvalues be reported?

Report Eigenvalue 1 = 3.195831, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

The first three observed roots exceed one and the fourth does not, which is consistent with a three-dimension solution. The retention decision is strengthened by parallel analysis rather than based on the Kaiser rule alone.

What Eigenvalues measures

Quantity estimated in this analysis

Interpretation that is not permitted

When to use Eigenvalues

Research question answered

Nearest methods that answer different questions

Real data used for Eigenvalues

Eigenvalues assumptions and design requirements

1. The matrix is symmetric and based on the intended variables

2. Variables are standardized when a correlation matrix is used

3. The eigenvalues are sorted consistently

4. Missing data and correlation type are fixed

5. PCA roots are not mislabeled common-factor roots

6. Rounding is not used before retention comparisons

Eigenvalues hypotheses or decision rule

Statistical question

Decision for the worked analysis

Eigenvalues formula and worked substitution

Symbol and denominator control

Full-precision substitution

Step-by-step Eigenvalues calculation

Establish the analytical object

Reconstruct the first required quantity

Verify the companion quantity

Apply the decision rule

Inspect local evidence

Reconcile and report

Eigenvalues results and interpretation

Primary result

Why the result is internally coherent

Eigenvalues in Python

Eigenvalues in R

Eigenvalues in SPSS or AMOS

Eigenvalues in Excel

Eigenvalues charts and visual diagnostics

01 Eigenvalues Primary Metrics

02 Eigenvalues Ordered Eigenvalues

03 Eigenvalues Eigenvectors

04 Eigenvalues Correlation Matrix

05 Eigenvalues Verified Result Summary

01 Eigenvalues Primary Metrics

02 Eigenvalues Ordered Eigenvalues

03 Eigenvalues Eigenvectors

04 Eigenvalues Correlation Matrix

05 Eigenvalues Verified Result Summary

Eigenvalues verification and sensitivity analysis

1. Verify that the nine roots sum to nine for a correlation-matrix PCA

2. Recalculate eigenpairs from the same variable order

3. Compare the third and fourth roots with simulated roots

4. Inspect the scree elbow around the retained boundary

5. Avoid interpreting eigenvector sign as substantive direction

6. Separate component variance from common-factor variance

Eigenvalues compared with related methods

Scree Plot

Parallel Analysis

Communalities

How to report Eigenvalues

Results paragraph

Settings that must accompany the result

Verification actions retained in the record

Eigenvalues decision scenarios

Boundary-case interpretation: Verify that the nine roots sum to nine for a correlation-matrix PCA

Input-definition sensitivity: Recalculate eigenpairs from the same variable order

Software-definition reconciliation: Compare the third and fourth roots with simulated roots

Local-chart conflict: Inspect the scree elbow around the retained boundary

Alternative-method challenge: Avoid interpreting eigenvector sign as substantive direction

Replication and reporting decision: Separate component variance from common-factor variance

Boundary-case interpretation: Verify that the nine roots sum to nine for a correlation-matrix PCA

Input-definition sensitivity: Recalculate eigenpairs from the same variable order

Software-definition reconciliation: Compare the third and fourth roots with simulated roots

Local-chart conflict: Inspect the scree elbow around the retained boundary

Eigenvalues downloads and reproducibility files

Eigenvalues frequently asked questions

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