midterm18335fall06solutions

midterm18335fall06solutions - MIT 18.335 Fall 2006 Midterm...

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Unformatted text preview: MIT 18.335, Fall 2006: Midterm, Solutions November 9, 2006 Name: • Do all of the 8 problems • Justify your answers • Exam time 90 minutes Grading 1 / 10 2 / 10 3 / 10 4 / 10 5 / 15 6 / 15 7 / 15 8 / 15 /100 1 1. (10 points) (a) Give the definition of a norm. (b) Consider a norm defined for a matrix A ∈ C m × n by k A k = max i,j ( | a ij | ) . Is this a valid norm? Prove your statement. (c) Show by a (simple) counter-example that this norm in general does not satisfy the bound on matrix products, k AB k ≤ k A k · k B k . Solution: (a) A norm is a function that assigns a real scalar to each x such that (1) k x k ≥ 0, and k x k = 0 only if x = 0 (2) k x + y k ≤ k x k + k y k (3) k αx k = | α |k x k (b) Yes, it is a valid norm: (1) k A k ≥ 0 because of abs function, and k A k = 0 only if all a ij = 0 (2) k A + B k = max i,j ( | a ij + b ij | ) ≤ max i,j ( | a ij | + | b ij | ) ≤ max i,j ( | a ij | ) + max i,j ( | b ij | ) = k A k + k B k (3) k αA k = max i,j ( | αa ij | ) = | α | max i,j ( | a ij | ) = | α |k A k (c) For example A = B = 1 1 1 1 = ⇒ k A k = k B k = 1 but k AB k = 2. 2 2. (10 points) Let A and X be m-by- m matrices with X nonsingular, and consider B = X- 1 AX . (a) Show that A and B have the same characteristic polynomial, and therefore the same eigenvalues. (b) If x is an eigenvector of A with eigenvalue λ , what is the correspond- ing eigenvector of B ? Solution: (a) p B ( z ) = det( zI- X- 1 AX ) = det( X- 1 ( zI- A ) X ) = det( X- 1 ) det( zI- A ) det( X ) = det( zI- A ) = p A ( z ). (b) The corresponding eigenvector is y = X- 1 x , since By = X- 1 AXX- 1 x = X- 1 Ax = X- 1 λx = λy....
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midterm18335fall06solutions - MIT 18.335 Fall 2006 Midterm...

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