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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)
Sâ€“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(Sâ€“1 A S) = det(A Sâ€“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 1Ã— 4 â€“ 2Ã— 3 = â€“2. has determinant 5.2 Ã— (â€“0.2) â€“ 0.2 Ã— 4.8 = â€“2. The trace of The trace of 32 is 1+4=5.
5.2 â€“ 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 â€œDistinct eigenvaluesâ...
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This document was uploaded on 09/16/2013.
 Spring '13

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