Eigenvalues

Eigenvalues, also known as characteristic roots or latent roots, are a special set of scalars related to MATRIX equations and are an important mathematical concept in a number of statistical methods including FACTOR ANALYSIS and PRINCIPAL COMPONENTS ANALYSIS.

Let A be a k × k square matrix of complex numbers; a scalar λ that belongs to the set of complex numbers is said to be an eigenvalue of A if there exists a nonzero k × 1 column vector X such that

This column vector X is known as the eigenvector of A. Because it is positioned to the right of A, it is called a “right eigenvector.” A row eigenvector that is positioned to the left of A is called a “left eigenvector.” Each eigenvalue is associated with a pair of eigenvectors—a left and a right eigenvector. The decomposition of A into eigenvalues and eigenvectors is known as eigen decomposition. The equation above also can be expressed more compactly as

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where I is the identity matrix. When an eigenvalue is distinct from all other eigenvalues, its eigenvector is unique. As an example of the application of eigenvalues, an eigenvalue in a dimension from a principal components analysis measures the goodness of fit and gives the proportion of variance in the original variables accounted for by the principal component.