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MAT1341-L17-Eigen0

# MAT1341-L17-Eigen0 - DIAGONALIZATION JOSE MALAGON-LOPEZ...

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DIAGONALIZATION JOS ´ E MALAG ´ ON-L ´ OPEZ Motivated by the fact that it is relatively easier to answer questions about diagonal matrices, in this lecture we deal with the problem of determining which matrices are “similar” to a diagonal matrix. We will see that this problem is closely related to the problem of finding vectors in R n for which Avectorv = λvectorv , for some λ . Diagonalizable Matrices Definition 1. Let A and B two n × n matrices. We say that A and B are similar if there is an invertible matrix Q such that A = Q 1 BQ. Theorem 2. If A and B are two similar n × n matrices, then det( A ) = det( B ) . Proof . If A and B are similar, by definition there is an invertible matrix Q such that A = Q 1 BQ . Then det( A ) = det ( Q 1 BQ ) = det ( Q 1 ) det( B ) det( Q ) = det ( Q 1 ) det( Q ) det( B ) = det ( Q 1 Q ) det( B ) = det ( I n ) det( B ) = det( B ) . Q.E.D. 1

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Definition 3. A square matrix is called diagonalizable if it is similar to a diagonal matrix. The problem of determining if a square matrix is diagonalizable, and if it is, how to obtain the expression of the form A = Q 1 DQ , where D is a diagonal matrix, can be solved using the remark below. Remark 4 . By definition, an n × n matrix A A = a 11 a 12 a 13 · · · a 1 n a 21 a 22 a 23 · · · a 2 n a 31 a 32 a 33 · · · a 3 n . . . . . . . . . a n 1 a n 2 a n 3 · · · a nn is diagonalizable if and only if there is an invertible matrix Q and a diagonal matrix D such that AQ = QD . Assume D = λ 1 0 0 · · · 0 0 λ 2 0 · · · 0 0 0 λ 3 · · · 0 . . . . . . . . . 0 0 0 · · · λ n with λ 1 , . . ., λ n real numbers not necessarily different. To say that Q is an invertible matrix is equivalent to saying that Col( Q ) = R n . Thus, the columns of Q consist of n linearly independent vectors in R n . Assume Q = q 11 q 12 q 13 · · · q 1 n q 21 q 22 q 23 · · · q 2 n q 31 q 32 q 33 · · · q 3 n . . . . . . . . . q n 1 q n 2 q n 3 · · · q nn 2
With these notations, the equality AQ = QD reads a 11 a 12 a 13 · · · a 1 n a 21 a 22 a 23 · · · a 2 n a 31 a 32 a 33 · · · a 3 n . . . . . . . . . a n 1 a n 2 a n 3 · · · a nn q 11 q 12 q 13 · · · q 1 n q 21 q 22 q 23 · · · q 2 n q 31 q 32 q 33 · · · q 3 n . . . . . . . . . q n 1 q n 2 q n 3 · · · q nn = q 11 q 12 q 13 · · · q 1 n q 21 q 22 q 23 · · · q 2 n q 31 q 32 q 33 · · · q 3 n . . . . . . . . . q n 1 q n 2 q n 3 · · · q nn λ 1 0 0 · · · 0 0 λ 2 0 · · · 0 0 0 λ 3 · · · 0 . . . . . . . . . 0 0 0 · · · λ n By definition of product of matrices, the i th column of the product AQ = QD is given as a 11 a 12 a 13 · · · a 1 n a 21 a 22 a 23 · · · a 2 n a 31 a 32 a 33 · · · a 3 n .

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MAT1341-L17-Eigen0 - DIAGONALIZATION JOSE MALAGON-LOPEZ...

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