The matrix P is nonsingular if and only if the n eigenvectors are linearly independent. In this case,

and we say that A is diagonalizable by the similarity transformation P ~ XAP. If >4 is a normal matrix, we can choose n eigenvectors that are orthonormal, or simultaneously orthogonal and normalized:

When the eigenvectors are orthonormal, P is a unitary matrix and A is diagonalizable by the unitary transformation A=P*AP. Any square matrix can be brought into Jordan canonical form [Hoffman and Kunze, 1961] by a similarity transformation


where the matrix J has the eigenvalues of A on the main diagonal and all zeros below the main diagonal. It follows from Eqs. (C-71), (C-59), and (C-60) that the trace of A is equal to the sum of its eigenvalues, and the determinant of A is equal to the product of its eigenvalues; i.e.,

Many algorithms exist for finding eigenvalues and eigenvectors of matrices, several of which are discussed by Carnahan, et al., [1969] and by Stewart [1973]. Using Eq. (C-61), we can see that the eigenvalues of Hermitian matrices are real numbers and the eigenvalues of unitary matrices are complex numbers with absolute value unity. Because the characteristic equation of a real matrix is a polynomial equation with real coefficients, the eigenvalues of a real matrix must either be real or must occur in complex conjugate pairs.

The case of a real orthogonal matrix deserves special attention. Because such a matrix is both real and unitary, the only possible eigenvalues are +1, -1, and complex conjugate pairs with absolute value unity. It follows that the determinant of a real orthogonal matrix is (- l)ffl where m is the multiplicity of the root A= -1 of the characteristic equation. A proper real orthogonal matrix must have an even number of roots at A= — 1, and thus an even number for all A ^ 1, because complex roots occur in conjugate pairs. Thus, an nXn proper real orthogonal matrix with n odd must have at least one eigenvector with eigenvalue +1. This is the basis of Euler's Theorem, discussed in Section 12.1.

It is also of interest to establish that the eigenvectors of a real symmetric mat/ix can be chosen to be real. The complex conjugate of the eigenvector equation, Eq. (C-63), is AX* =AX*, because both A and A are real. Thus, X* is an eigenvector of A with the same eigenvalue as X. Now, either X = X*, in which case the desired result is obtained, or X ^ X*. In the latter case, we can replace X and X* by the linear combinations X+X* and i(X —X*), which are real eigenvectors corresponding to the eigenvalue A. Thus, we can always find a real orthogonal matrix P to diagonalize a real symmetric matrix A by Eq. (C-69).

C.7 Functions of Matrices

Let f(x) be any function of a variable x, for example, sinx or expx. We want to give a meaning to f(A/), where M is a square matrix. If f(x) has a power series expansion about x=0, f(x)= 2 fl„x-0

then we can formally (i.e., ignoring questions of convergence) define t(M) by

with the same coefficients a„. It is clear that f(A/) is a square matrix of the same order as M. If M is a diagonalizable matrix, then by Eq. (C-69),

where P is the matrix of eigenvectors defined by Eq. (C-66), and A is the diagonal matrix of eigenvalues. Then,

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