Lecturenotes28-30March2011

Lecturenotes28-30Mar - Section 5.2 Diagonalization Definition If A and B are square matrices then we say that B is similar to A if there is an

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Unformatted text preview: Section 5.2 Diagonalization Definition: If A and B are square matrices, then we say that B is similar to A if there is an invertible matrix P such that B = P- 1 AP. Facts: 1. A and B have the same determinant 2. A is invertible if and only if B is invertible 3. A and B have the same rank 4. A and B have the same nullity 5. A and B have the same trace 6. A and B have the same charactesitic polynomial 7. A and B have the same eigenvalues 8. If λ is an eigenvalue of A and hence B, then the eigenspaces of A corresponding to λ and the eigenspace of B corresponding to λ have the same dimension Note: If B as described P- 1 AP , is a diagonal matrix then finding eigenvalues and eigenspace of A will be much easier. Definition: A square matrix A is said to be diagonalizable if it is similar to some di- agonal matrix; that is, if there exists an invertible matrix P such that P- 1 AP is diagonal. In this case the matrix P is said to diagonalize A....
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Lecturenotes28-30Mar - Section 5.2 Diagonalization Definition If A and B are square matrices then we say that B is similar to A if there is an

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