Page 827 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(2)

The reader should verify that matrix 2 is the inverse of matrix 1 by showing that .

Charles Hermite (1822–1901) was a French mathematician who made fundamental contributions to algebra, matrix theory,
and various branches of analysis. He is noted for using integrals to solve a general fifth-degree polynomial equation. He also
proved that the number (the base for natural logarithms) is a transcendental number—that is, a number that is not the root
of any polynomial equation with rational coefficients.

Recall that a square matrix A with real entries is called orthogonally diagonalizable if there is an orthogonal matrix such that
is diagonal. For complex matrices we have an analogous concept.

DEFINITION

A square matrix A with complex entries is called unitarily diagonalizable if there is a unitary such that
is diagonal; the matrix is said to unitarily diagonalize .

We have two questions to consider:

Which matrices are unitarily diagonalizable?

How do we find a unitary matrix to carry out the diagonalization?

Before pursuing these questions, we note that our earlier definitions of the terms eigenvector, eigenvalue, eigenspace,
characteristic equation, and characteristic polynomial carry over without change to complex vector spaces.

Hermitian Matrices