Similarity - ME 210 Applied Mathematics for Mechanical...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Similar Matrices, Similarity Transformation and Diagonalization Two square matrices [A] and [B] of the same size n are called similar if there exits a non-singular [P] matrix of size n such that the matrix [B] is obtained by means of a so-called similarity transformation defined as [B] = [P] -1 [A] [P] or [P] [B] = [A] [P] or [A] = [P] [B] [P] -1 or alternatively [B] = [Q] [A] [Q] -1 or [B] [Q] = [Q] [A] or [A] = [Q] -1 [B] [Q] where [Q] = [P] -1
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Determine whether the following [A] and [B] matrices are similar or not. 1 2 A = -1 4    2 0 B = -2 3 The similarity transformation requires that [P] [B] = [A] [P] Define [P] as   11 12 21 22 p p P = p         11 12 11 12 21 22 21 22 p p 2 0 1 2 P B = = = A P p -2 3 -1 3 p 11 12 21 12 22 11 21 22 12 22 2 2 2 2 0 0 = 2 2 0 0 p p p p p p p p p p
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 This gives these four equations in four unknowns, but only two are linearly independent 11 12 21 11 21 22 12 22 12 22 p - 2 p - 2 p = 0 2 p - 2 p = 0 p - p = 0 11 12 21 12 22 Define, arbitrarily, p 22 = c ≠ 0 and p 11 = 0, then   11 12 21 22 p p
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Similarity - ME 210 Applied Mathematics for Mechanical...

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