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Unformatted text preview: Friday March 27 Lecture 33 : Diagonalizable matrices . (Refers to 6..1, 6.2 and 6.3) Expectations: 1. Define diagonalizable n n matrix A . 2. Recognize that a matrix A is diagonalizable if the matrix A has n distinct eigenvalues. 3. Diagonalize simple 2 by 2 or 3 by 3 matrices. 33.1 Definition A matrix A is said to be diagonalizable if there exists an invertible matrix P such that the matrix D = P 1 AP is a diagonal matrix, that is, a matrix D with zero entries except possibly on the main diagonal, (equivalently A = XDX-1 ). 33.1.1 Remarks Recall the definition of similar matrices: Two matrices A and C are said to be similar if they satisfy the property C = B-1 AB for some invertible matrix B . Thus A is said to be diagonalizable if A is similar to a diagonal matrix D . We showed earlier that if C = B 1 AB then C and A have the same eigenvalues. We also showed that the eigenvalues of a triangular matrix are those numbers which are on the diagonal. So if D is a diagonal matrix which is similar to the matrix A then the eigenvalues of A are those numbers sitting the diagonal of D . 33.2 Definition Finding matrices P and P 1 so that P 1 AP is a diagonal matrix is a process called diagonalizing a matrix A . Of course, we can only diagonalize a matrix A if the matrix A is diagonalizable. Note that some authors use the expression P diagonalizes A if P 1 AP is a diagonal matrix. Our objective is twofold: 1) Determine when a matrix A is diagonalizable. 2) If A is diagonalizable what is an efficient way of diagonalizing A . 33.3 Theorem Suppose A is an matrix n n is a matrix with n distinct eigenvalues. Then the n n matrix X whose columns are n eigenvectors who are respectively associated to the n distinct eigenvalues of a matrix A is invertible....
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This note was uploaded on 06/10/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.
- Winter '08