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Properties of condition numbers : Orthogonal Matrices and Eigenvalues

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Please prove the properties of condition numbers attached to this message. Refer to definitions/theorems you used. Also, if you want, have a look at the second file attached, since I believe that you can refer to the previous properties to do 6 to 10.

7. For any orthogonal matrix Q,
i2(QA) = k2(AQ) = k2(A)
8. If D= diag(d1,...,d) then
.....
9. If ... is the largest eigenvalue of A'A and 'm is its smallest eigenvalue, then
....
10. c2(A) = 1 if and only if A is a multiple of an orthogonal matrix.

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Orthogonal Matrices and Eigenvalues are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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The definition of the condition number of is . So we always consider that is non-singular.
Proof:
1. Since , then by definition, we have

Thus we get .
2. We know, . I claim that .
For any non-zero , we consider . We have

Thus, .
On the other side, for any , we have
.
This inequality holds for any and ...

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