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7.2 In this section we shall be concerned with the problem of finding a basis for
DIAGONALIZATION that consists of eigenvectors of a given matrix A. Such bases can be
used to study geometric properties of A and to simplify various numerical
computations involving A. These bases are also of physical significance in a
wide variety of applications, some of which will be considered later in this text.
The Matrix Diagonalization Problem
Our first objective in this section is to show that the following two problems, which on the surface seem quite different, are
actually equivalent.
The Eigenvector Problem Given an matrix A, does there exist a basis for consisting of eigenvectors of A?
The Diagonalization Problem (Matrix Form) Given an matrix A, does there exist an invertible matrix P such that
is a diagonal matrix?
The latter problem suggests the following terminology.
DEFINITION
A square matrix A is called diagonalizable if there is an invertible matrix P such that is a diagonal matrix; the matrix
P is said to diagonalize A.
The following theorem shows that the eigenvector problem and the diagonalization problem are equivalent.
THEOREM 7.2.1
If A is an matrix, then the following are equivalent.
(a) A is diagonalizable.
(b) A has n linearly independent eigenvectors.
Proof Since A is assumed diagonalizable, there is an invertible matrix
