12.2 Properties of Eigenstructures

Knowing about the properties of eigenstructures will pay off when you encounter methods whose solutions involve an eigenvalue decomposition.

Because all the following properties have the same premise, I prefer to state it here: suppose that matrix \(\mathbf{A}\) has an eigenvalue \(\lambda\) and an associated eigenvector \(\mathbf{v}\):

\[ \mathbf{Av} = \lambda \mathbf{v} \]

Prop. 1

The matrix \(b \mathbf{A}\), where \(b\) is an arbitrary scalar, has \(b \lambda\) as an eigenvalue, with \(\mathbf{v}\) as the associated eigenvector.

Prop. 2

The matrix \(\mathbf{B} = \mathbf{A} + c \mathbf{I}\), where \(c\) is an arbitrary scalar, has \((\lambda + c)\) as an eigenvalue, with \(\mathbf{v}\) as the associated eigenvector.

Prop. 3

The matrix \(\mathbf{A}^m\), where \(m\) is any positive integer, has \(\lambda^m\) as an eigenvalue, with \(\mathbf{v}\) as the associated eigenvector.

In other words, if \(\mathbf{v}\) is an eigenvector of \(\mathbf{A}\), then \(\mathbf{v}\) is also an eigenvector of \(\mathbf{A}^2\), \(\mathbf{A}^3\), and so on. Likewise, \(\lambda^2\) is an eigenvalue of \(\mathbf{A}^2\), \(\lambda^3\) is an eigenvalue of \(\mathbf{A}^3\), etc.

Prop. 4

The matrix \(\mathbf{A}^{-1}\), assuming that \(| \mathbf{A} | \neq 0\), has \(1 / \lambda\) as an eigenvalue, with \(\mathbf{v}\) as the associated eigenvector.

Also, when \(\mathbf{A}^{-1}\) exists, then \(\mathbf{A}^{-p}\) has the same eigenvectors as \(\mathbf{A}\) and \(\lambda_{k}^{n}\) is the \(k\)-th eigenvalue.

In general, \(\mathbf{A}^{-1} \mathbf{V} = \mathbf{V} \mathbf{\Lambda}^{-1}\).

Prop. 5

If \(\mathbf{A}\) is symmetric, then the eigenvectors of \(\mathbf{A}\) are also eigenvectors of \(\mathbf{A^\mathsf{T} A}\).

Prop. 6

The sum of the eigenvalues of a matrix is equal to the sum of the diagonal elements of the matrix (the trace of the matrix).

\[ tr(\mathbf{A}) = \sum_{k=1}^{n} \lambda_k \]

Prop. 7

The product of the eigenvalues of \(\mathbf{A}\) equals the determinant of \(\mathbf{A}\):

\[ \prod_{k=1}^{n} \lambda_k = | \mathbf{A} | \]

Prop. 8

If \(\mathbf{A}\), a matrix of order \(n \times n\), has rank \(r\), then \(\mathbf{A}\) has \(n - r\) eigenvalues equal to zero.

Prop. 9

If a symmetric matrix \(\mathbf{A}\) can be written as the product \(\mathbf{A} = \mathbf{V \Lambda V^\mathsf{T}}\), where \(\mathbf{\Lambda}\) is diagonal with all entries nonnegative and \(\mathbf{V}\) is an orthogonal matrix of eigenvectors, then:

\[ \mathbf{A}^{1/2} = \mathbf{V \Lambda}^{1/2} \mathbf{V} \]

and it is the case that \(\mathbf{A}^{1/2} \mathbf{A}^{1/2} = \mathbf{A}\).

Prop. 10

If a symmetric matrix \(\mathbf{A}^{-1}\) can be written as the product \(\mathbf{A}^{-1} = \mathbf{V \Lambda}^{-1} \mathbf{V^\mathsf{T}}\), where \(\mathbf{\Lambda}^{-1}\) is diagonal with all entries nonnegative and \(\mathbf{V}\) is an orthogonal matrix of eigenvectors, then:

\[ \mathbf{A}^{-1/2} = \mathbf{V \Lambda}^{-1/2} \mathbf{V} \]

and it is the case that \(\mathbf{A}^{-1/2} \mathbf{A}^{-1/2} = \mathbf{A}^{-1}\).

12.2.1 Symmetric Matrices

The following table lists characteristics of eigenvalues depending on the type of symmetric matrices.

Symmetric matrix	Eigenvalues
All elements of \(\mathbf{A}\) are real	All eigenvalues are real
\(\mathbf{A}\) is positive definite	All eigenvalues are positive
\(\mathbf{A}\) is positive semidefinite and of rank \(r\)	There are \(r\) positive and \(p-r\) null eigenvalues
\(\mathbf{A}\) is negative semidefinite and of rank \(r\)	There are \(r\) negative and \(p-r\) null eigenvalues
\(\mathbf{A}\) is indefinite and of rank \(r\)	There are \(r\) non-null and \(n-r\) null eigenvalues
\(\mathbf{A}\) is diagonal	The diagonal elements are the eigenvalues