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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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- Winter '08
- Matrices, Diagonal matrix, distinct eigenvalues