Spectral theorem1 suppose that a is a symmetric n n

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Unformatted text preview: metric matrices Before proceeding further, we need to review and extend some notions from vectors and matrices (linear algebra), which the student should have studied in an earlier course. In particular, we will need the amazing fact that the eigenvalue-eigenvector problem for an n × n matrix A simplifies considerably when the matrix is symmetric. An n × n matrix A is said to be symmetric if it is equal to its transpose AT . Examples of symmetric matrices include 3−λ 6 5 abc 13 6 b d e . 1−λ 0 , and 31 5 0 8−λ cef Alternatively, we could say that an n × n matrix A is symmetric if and only if x · (Ay) = (Ax) · y. (2.1) for every choice of n-vectors x and y. Indeed, since x · y = xT y, equation (2.1) can be rewritten in the form xT Ay = (Ax)T y = xT AT y, which holds for all x and y if and only if A = AT . On the other hand, an n × n real matrix B is orthogonal if its transpose is equal to its inverse, B T = B −1 . Alternatively, an n × n matrix B = (b1 b2 · · · bn ) 29 is orth...
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This document was uploaded on 01/12/2014.

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