What Is an Eigenvector? How to Read Eigenvalues and Eigenvectors in SPSS

What Is an Eigenvector

You open the output from your exploratory factor analysis, or EFA, and see terms such as Eigenvalue, eigenvector, Total Variance Explained, and Rotated Component Matrix. These concepts are connected to the way the software reduces many observed variables to a smaller number of more meaningful factors.

Mathematically, an eigenvector is a special vector of a matrix. When the matrix acts on this vector, the vector keeps the same direction while its magnitude changes. The coefficient that describes this change is called the eigenvalue. It can be written as:

A v = λ v

Here, A is the data matrix or correlation matrix, v is the eigenvector, and λ is the eigenvalue. When you run EFA, you usually do not calculate eigenvectors by hand. SPSS performs this calculation to identify the main directions of variation in the data, then presents the eigenvalues in the Total Variance Explained table.

The terminology is easy to mix up. An eigenvector is the vector used in the mathematical transformation. An eigenvalue is the value attached to that vector and is commonly used to assess how much variance a factor explains. When you interpret EFA output, you usually need to discuss the eigenvalue and the percentage of variance extracted. The eigenvector generally does not appear as a separate conclusion column in the final results table.

If you are working through the full scale-development process, you can also read what is a scale to distinguish a scale, latent variable, and observed variable before running EFA.

The Meaning of an Eigenvector in Quantitative Research

In quantitative research, an eigenvector helps identify the direction in which observed variables combine within the data space. Each observed variable may vary together with other variables. The correlation matrix records those relationships, while the eigenvector provides directions in which the data show substantial variation.

When you run EFA, your practical goal is to check whether the observed variables group into factors that fit the theoretical model. The eigenvalue helps you decide how many factors to retain. The eigenvector belongs to the calculation behind the scenes. It helps create the factors or components before SPSS displays factor loadings in the Component Matrix or Rotated Component Matrix.

For example, a questionnaire may contain 20 observed variables measuring service quality, convenience, trust, and continuance intention. If EFA shows four clear groups, you can continue by checking scale reliability and running a regression model or PLS-SEM. If some variables load on several groups, you need to review the item wording, factor loadings, loading differences, and theoretical basis.

The eigenvector also explains why EFA is more than clicking through a menu. The software transforms the correlation matrix into new axes. These axes are selected to describe a large share of the variation in the data. After factor rotation, the direction of the axes may change so that the factor structure is easier to interpret. You still need to read the total explained variance according to the extraction method and rotation method you selected.

You also need to keep EFA separate from other data checks. An eigenvector does not tell you whether your sample contains an outlier. If a few rows contain unusual values, check outliers in SPSS before concluding that the factor structure is defective. Similarly, an eigenvector is not an indicator for assessing a moderator or mediator. Those topics use their own tests, such as moderating and mediator.

What Eigenvector Value Is Acceptable

There is no independent cutoff such as “from what value is an eigenvector acceptable” that you can use to keep or remove a variable. In EFA output, the decision usually depends on the eigenvalue, the number of extracted factors, the percentage of variance extracted, factor loadings, and the fit of the measurement model.

If your lecturer asks, “What eigenvector value is acceptable,” first identify which indicator they mean in the SPSS table. The table below separates the criteria that are often incorrectly called eigenvectors.

An eigenvalue greater than 1 is a starting rule, not the only evidence you need to determine the number of factors. This rule is presented in (Kaiser, 1960). You should also examine the Scree Plot, the theoretical basis, and the meaning of each item group. A factor with an eigenvalue just above 1 but with items that are difficult to explain still needs to be discussed with your lecturer.

Factor loading is the result you can see more directly after extracting and rotating the factors. A level of 0.5 or above is commonly used as a reference threshold in EFA (Hair et al., 2010). Some sources may accept a lower threshold in exploratory research, depending on the context. If you do that, state your reason instead of retaining an item simply to make the table look better.

How to Read Eigenvectors in the Output

In SPSS, the eigenvector does not necessarily appear under the exact name Eigenvector in the table you use to write Chapter 4. Read the output as a sequence of tables: KMO and Bartlett's Test, Total Variance Explained, Scree Plot, Rotated Component Matrix, and, in some cases, Component Transformation Matrix.

Reading KMO and Bartlett's Test

KMO indicates whether the data are suitable for factor analysis. Bartlett's Test of Sphericity tests whether the correlation matrix differs from an identity matrix. In the illustrative figures below, KMO is 0.812 and Bartlett's Sig. is 0.000, so the data are suitable for continuing with EFA under the reference thresholds stated above.

Reading Total Variance Explained

In the Total Variance Explained table, the Initial Eigenvalues column shows the eigenvalue of each component before reduction. Extraction Sums of Squared Loadings shows the part retained after the extraction method. If you use rotation, Rotation Sums of Squared Loadings helps you see how variance is distributed among the factors after rotation.

The table below is illustrative output, not the result of a specific study.

In this example, the first three components have eigenvalues greater than 1. The total variance extracted by the three components is 55.953%, above the 50% reference level (Hair et al., 2010). You still need to examine the Rotated Component Matrix to see how the observed variables are distributed across the three factors.

Reading Rotated Component Matrix

Rotated Component Matrix displays the loading of each variable on the factors after rotation. An illustrative table may look like this:

The figures in this table are factor loadings, not eigenvectors. Check which factor has the highest loading for each variable, whether there is substantial cross-loading, and whether the group fits the theoretical definition. Then name the factor based on the item content. Do not name it simply because Component 1 has the highest eigenvalue.

Reading Scree Plot and Component Transformation Matrix

Scree Plot helps you observe the point at which the eigenvalue curve bends. If the line falls sharply across the first few components and then becomes flatter, the number of factors before the bend is useful for comparison with the rule based on eigenvalues greater than 1.

Component Transformation Matrix commonly appears when you use a rotation such as Varimax. This matrix describes the transformation between the components before and after rotation. You do not need to report the numbers in this table as factor loadings or eigenvalues.

What to Do When the Eigenvector Does Not Meet the Criteria

First, identify whether “does not meet the criteria” refers to the eigenvalue, explained variance, or factor loading. There is no button that fixes an eigenvector. Your response should move from data checking to scale checking, in an order that you can explain in the thesis.

Check the Number of Variables and Data Types

Open Variable View and Data View. Check whether the variables use the same direction of measurement, whether reverse-coded items have been reverse coded, and whether missing data have been entered as a false value. An untreated reverse-coded item can reduce the correlations among items and make the factor structure difficult to interpret.

Review KMO, Bartlett, and the Correlation Matrix

If KMO is low, the data do not yet show enough correlation for a suitable EFA. If Bartlett is not statistically significant, the variables may not be related strongly enough. At that point, do not randomly delete a few variables simply to force KMO upward. Recheck the scale group, data entry, and sample size.

Remove a Variable With a Rationale and Run Again

If a variable has a low factor loading, high cross-loading, or content that does not fit the factor, you may consider removing it. Record each removal round: which variable was removed, why it was removed, how the indicators changed, and whether the new structure still fits the theoretical model. EFA rarely looks completely clean on the first run, but every removal needs an argument.

Do Not Change the Method Just to Get a Better Table

PCA and common factor analysis answer different questions. Orthogonal and Oblique rotation also produce different forms of interpretation. Select the option that fits your model and report it clearly in the thesis. If your lecturer asks you to confirm the measurement model with SmartPLS or AMOS, move to that tool instead of endlessly adjusting EFA in SPSS.

Distinguishing Eigenvector, Eigenvalue, and Factor Loading

These three terms are related, but you cannot use them interchangeably. An eigenvector is a special vector of a matrix, an eigenvalue indicates the amount of variation associated with that vector, and a factor loading indicates the relationship between an observed variable and the interpreted factor.

The eigenvalue is commonly used to decide which factors may be retained. Factor loading is used to assess which factor an observed variable belongs to. The eigenvector belongs to the mathematical stage behind matrix decomposition, so you normally do not use it to conclude that a variable is acceptable or unacceptable.

You should also avoid confusing an eigenvalue with R² in regression. An eigenvalue relates to the variance structure of a matrix in EFA or PCA. R² shows the proportion of variation in the dependent variable explained by the regression model. If you are reading a model with an interaction variable, also see what is a moderator so that you do not use the terms incorrectly.

Another common confusion is treating an eigenvector as a scale. A scale consists of observed variables designed to measure a construct. An eigenvector is not an observed variable and is not an item in a questionnaire. It is a mathematical component generated from the data matrix.

Common Errors

Calling an Eigenvalue an Eigenvector

This is the most common terminology error. When you write Chapter 4, use the correct table names, such as Initial Eigenvalues, Total Variance Explained, or Rotated Component Matrix. If you mean the value used to retain a factor, the appropriate term is usually eigenvalue.

Concluding Only From an Eigenvalue Greater Than 1

An eigenvalue greater than 1 is a reference rule. You still need to examine the Scree Plot, total variance extracted, factor loadings, and theoretical meaning. Retaining three factors simply because three eigenvalues are greater than 1, while the items lack a consistent meaning, will invite questions from the committee.

Reporting an Eigenvector as a Loading

The numbers in Rotated Component Matrix are factor loadings. Do not write that “the eigenvector of variable DV1 is 0.812” if 0.812 actually appears in the Component 1 column. That wording shows that you have misread the output table.

Deleting Variables Until the Indicators Look Good

Running EFA again across several rounds may be necessary, but repeatedly deleting variables without recording the reason weakens the credibility of the procedure. Keep the output file from every round and create a tracking table for the removed variables. If the result moves away from the theoretical model, discuss it before changing the model.

Using an Unsupported Threshold

In a thesis, each threshold needs an appropriate source. The rule that the eigenvalue must be greater than 1 can be cited as (Kaiser, 1960), KMO can be cited as (Kaiser, 1974), and factor loading from 0.5 can be cited as (Hair et al., 2010). You can also review articles in the Statistics category to compare how other SPSS tables are interpreted.

Frequently asked questions

What are eigenvectors and eigenvalues?

An eigenvector keeps the same direction when acted on by a matrix, while the eigenvalue is the coefficient that changes the vector's magnitude. In EFA, the eigenvalue is commonly used to assess how much variance a component or factor explains.

What eigenvector value is acceptable?

An eigenvector has no independent acceptance threshold used for conclusions in EFA. You need to examine the eigenvalue, factor loading, KMO, Bartlett's Test, and Total Variance Explained. The rule that the eigenvalue must be greater than 1 is a reference criterion presented in (Kaiser, 1960).

Should a factor with an eigenvalue below 1 be removed immediately?

Under the Kaiser rule, a component with an eigenvalue below 1 is generally not retained (Kaiser, 1960). However, you should also examine the Scree Plot and the theoretical basis. Do not remove or retain a factor based on one number when the scale structure cannot be explained clearly.

What should I do if SPSS does not display the eigenvector?

Open the Total Variance Explained, Scree Plot, and Rotated Component Matrix tables. SPSS usually presents the eigenvalue and factor loading in the tables needed for interpretation. The eigenvector may remain part of the internal calculation or appear in the component transformation table, depending on the extraction method.

Do I need to include the eigenvector in Chapter 4?

You usually report the EFA method, KMO, Bartlett's Test, the number of retained factors, eigenvalue, total variance extracted, and factor loading. Explain the eigenvector only when the topic or your lecturer requires the mathematical part of matrix decomposition.

How to write this in your thesis

You can adapt the following sentence: “The EFA results show that the data are suitable for factor analysis, with KMO = [value], Bartlett's Test with Sig. = [value], [number of factors] factors retained because their eigenvalues are greater than 1, total variance extracted reaching [value]%, and factor loadings ranging from [lowest value] to [highest value].” Now reopen your SPSS file, identify the correct Total Variance Explained and Rotated Component Matrix tables, and check every figure before adding it to Chapter 4. If you need to run the analysis on your own .sav or .csv file, M4 analysis by DoThesis is the relevant module.