1. Linear Algebra Chapter 6 Linear Algebra with Applications -Gareth Williams Br. Joel Baumeyer, F.S.C.

2. Definition: Eigenvalue & Eigenvector n The Characteristic Polynomial of A nxn : |A - I | n The Characteristic Equation of A nxn : |A - I | = 0

3. Theorem 6.1

4. Similar Matrices n Definition: Let A and B be square matrices of the same size. B is said to be similar to A if there exists an invertible matrix C such that B = C -1 AC. The transformation of the matrix A into the matrix B in this manner is called a similarity transformation.

5. Theorem 6.3 Theorem 6.2 is in the optional section 6.2 and not covered.

6. Diagonalizable Matrix n A square matrix A is said to be diagonalizable if there exists a matrix C such that D = C -1 AC is a diagonal matrix.

7. Theorem 6.4 n Let A be an n  n matrix. –(a) If A has n linearly independent eigenvectors, it is diagonalizable. The matrix C whose columns consist of n linearly independent eigenvectors can be used in a similarity transformation C -1 AC to give a diagonal matrix d. The diagonal elements of D will be the eigenvalues of A. –(b) If A is diagonalizable, then it has n linearly independent eigenvectors.

8. Powers of Diagonalizable Matrices n If A is similar to a diagonal mamatrix D under the transformation C -1 AC, then it can be shown that A k = CD k C -1. n Demonstration: D k = (C -1 AC) k = (C -1 AC) … (C -1 AC) = (C - 1 A k C) and reversing gives: A k = CD k C -1

9. Theorem 6.5 n Let a be an nxn symmetrix matrix. –a) All the eigenvalues of A are real numbers. –b) the dimension of an eigenspace of A is the multiplicity of the eigenvalue as a root of the characteristic equation. –c) The eigenspaces of A are orthogonal. –d)A has n linearly independent eigenvectors.

10. Orthogonally Diagonalizable n Definition: A square matrix A is said to be orthogonally diagonalizable if there exists an ortholgonal matrix C such that D = C t AC is a diagonal matrix.

11. Theorem 6.6 n Let a be a square matrix. A is orthog- onally diagonalizable if and only if it is a symmetric matrix.