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Unformatted text preview: matrix using Gauss elimination? 8. What is a norm? And a matrix norm? 9. The Jordan decomposition theorem states that given a matrix A ∈ R n × n , there exists a nonsingular matrix P ∈ R n × n such that P1 AP = J 1 . . . J 2 . . . . . . . . . . . . . . . . . . J r , 1 2 where the blocks J k are of the form J k = λ I = λ . . . λ . . . λ . . . . . . . . . . . . . . . . . . . . . λ , or J k = λ 1 . . . λ 1 . . . λ 1 . . . . . . . . . . . . . . . . . . . . . . . . λ 1 . . . λ , where λ is an eigenvalue of A . Prove by direct computation, that in either case lim m →∞ J k m = 0 ⇐⇒  λ  < 1 . Use the same proof to show that J k m is bounded if and only if: (a)  λ  ≤ 1 if the block is diagonal. (b)  λ  < 1 if the block is not diagonal....
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 Fall '08
 Ceniceros,H
 Math, Linear Algebra, Invertible matrix, Computational Complexity, JK M

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