SelfAssessment1

# SelfAssessment1 - matrix using Gauss elimination 8 What is...

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Self-Assessment 1 – Math 104B, Spring 2009 Answer the following questions without looking in the book. If you do not feel comfortable answering them, read the corresponding sections in the book, and then solve the problems, again without looking in the book. 1. What is the diﬀerence between a direct method and an iterative method for Ax = b ? 2. What is the computational complexity for Gauss elimination for a gen- eral matrix A ? What if the matrix is tridiagonal? 3. What is the computational complexity of computing the Choleskii de- composition for a symmetric, positive deﬁnite matrix A ? 4. What is the computational complexity of backward substitution for a general triangular matrix? 5. What is Crout’s algorithm? How does the complexity of Crout’s algo- rithm compare to Gauss elimination? 6. What is the computational complexity of computing a determinant from its deﬁnition? How can Gauss elimination be used to compute the determinant of a matrix? What would be the computational complexity in that case? 7. What is the computational complexity of computing the inverse of a

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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 P-1 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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SelfAssessment1 - matrix using Gauss elimination 8 What is...

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