MATH115 Lecture 31: lect115_31_f14
10 Oct 2015
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Friday, november 14 lecture 31 : orthogonal diagonalization. 31. 0 definition a square matrix a is symmetric if at = a. 31. 1 definition a square matrix y is called an orthogonal matrix if its column vectors form an orthonormal set of vectors. 31. 1. 1 example the matrix has orthogonal columns but cannot be called an orthogonal matrix since the norm of the column vectors is not 1. But we can normalize these column vectors to obtain an orthogonal matrix a1 : Here is an important and useful characterization of orthogonal matrices. 31. 2 proposition a square matrix y is an orthogonal matrix if and only if y 1 = y t. Furthermore y is orthogonal if and only if y 1 is orthogonal. Proof: follows from ptp = i if and only if p has a columns which form an orthonormal set. 31. 2. 1 proposition a matrix y is an orthogonal matrix if and only if y t is orthogonal.
