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Diagonalization of a square matrix.

A diagonal matrix is a square matrix whose only non zero entries are on the diagonal, i.e. A=(aij) is diagonal if, and only if, . If the diagonal entries are , we denote .

For example, is a diagonal matrix.

A square matrix A is diagonalizable if there exists an invertible matrix P such that P-1AP is a diagonal matrix.

Theorem 11.6.1   Consider the square matrix A of order n as the matrix of an endomorphism of w.r.t. the standard basis. If there exist a basis of composed only from eigenvectors of A, then A is diagonalizable.

The matrix P is the change of base matrix from the standard basis to the basis of eigenvectors.

Example 11.6.2   With the settings of  4.3, we have the eigenvectors and . They are linearly independent; as , they form a basis of V. Define the matrix ; then and .

Noah Dana-Picard
2001-02-26