ERG2013_lecture_24

We say that two n n matrix a and b are similar if we

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Unformatted text preview: similar. kshum Analogy with triangles Two triangles are congruent if the corresponding sides have the same length. Two triangles are similar if the corresponding angles are equal. Two matrices are equal if all corresponding entries are equal. Two square A and B matrices are similar if equal congruent similar 23 similar kshum Similar matrices represent the same linear system in two different coordinates 24 kshum The same system in two different coordinates Spiral source x '=A x +By y ' = Cx +Dy x '=A x +By y ' =Cx +Dy A =4 B=3 C=- 2 D=3 4 3 3 2 2 1 1 0 0 y 5 4 y 5 A = 6.8 B = 6.4 C = - 2.6 D = 0.2 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 x 25 0 x kshum 1 2 3 4 5 The same system in two different coordinates Jordan node x '=A x +By y ' =Cx + Dy x '=A x +By y ' = Cx +Dy A =- 3 B=1 C= 0 D=- 3 5 5 4 4 3 3 2 2 1 y 1 y A = - 3.5 B = 1 C = - 0.25 D = - 2.5 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -5 -5 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 5 0 1 2 3 4 5 x x In both cases, only one linearly independent eigenvector can be found. 26 kshum Triangle similarity is an equivalence relation Reflexive: Every triangle is similar to itself similar Symmetric: If triangle A is similar to triangle B, then B is similar to A. A similar B B similar A Transitive: If triangle A is similar to triangle B, and triangle B is similar to triangle C, then A is similar to C. A B 27 similar similar B A similar C kshum C Matrix similarity is an equivalence relation Reflexive: Every matrix is similar to itself Symmetric: If A is similar to B, then B is similar to A. Transitive: If A is similar to B, and B is similar to C, then A is similar to C. Suppose 28 kshum Similar triangles have the same angles 29 kshum Similar matrices have the same determinant Suppose that A and B are similar, i.e., by similarity det(MN) = det(M) det(N) det(MN) = det(M) det(N) S–1 is the inverse of S det(I) = 1 Note that in the third equality, the exchange of order of multiplication is allowed because it is a multiplication of two numbers. Multiplication of matrices is not commutative in general. We cannot write det(S–1 A S) = det(A S–1S) in the first line.. 30 kshum Similar matrices have the same trace Trace of a matrix = sum of diagonal entries Proof: exercise 31 kshum Example has determinant 1× 4 – 2× 3 = –2. has determinant 5.2 × (–0.2) – 0.2 × 4.8 = –2. The trace of The trace of 32 is 1+4=5. 5.2 – 0.2 = 5. kshum Similar matrices have the characteristic polynomial Characteristic polynomial of A Suppose Characteristic polynomial of B Since eigenvalues are the roots of characteristic polynomial, similar matrices have the same eigenvalues. 33 kshum Analogy similar similar Same angles 34 Both matrices have the same eigenvalues kshum Theorem (theorem 1 in Kreyszig 8.4) If eigenvalues are all distinct, then the matrix is similar to a diagonal matrices, whose diagonal entries are precisely Proof can be found in p.349 the eigenvalues. Example: Eigenvalue = 0, 3, 9 “Distinct eigenvalues...
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This document was uploaded on 09/16/2013.

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