Linear Algebra Sample Quals

Linear Algebra Sample Quals - , T = 4-1-1 . . . . . . . . ....

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1. Let A be an n × n matrix such that k A k < 1 for some natural norm k · k . Prove that I - A is nonsingular, and that 1 1 + k A k ≤ k ( I - A ) - 1 k ≤ 1 1 - k A k . 2. Let A be an m × n matrix with rank( A ) < r , where r < min { m,n } . Explain how Householder transformations and permutation matrices can be used to compute the pseudo-inverse of A . 3. Let u be an n -vector, and consider the n × n matrix A = I + uu T . (a) Describe the eigenvalues and eigenvectors of A . (b) Use your answer to part (a) to obtain the SVD of A . (c) Compute A - 1 . 4. Consider the Helmholtz equation on the unit rectangle R , Δ u + σ ( x,y ) u = f, ( x,y ) R, u = g, ( x,y ) ∂R, where σ ( x,y ) > 0 on R . We discretize Δ on an ( N +1) × ( N +1) grid using an N 2 × N 2 matrix A , where A = - T I I . . . . . . . . . . . . I I - T
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Unformatted text preview: , T = 4-1-1 . . . . . . . . . . . .-1-1 4 . We then solve the discretized equation using the iteration A u ( k +1) + h 2 u ( k ) = f where = 11 . . . NN , ij = ( x i ,y j ) . 1 (a) Show that as h 0, the rate of convergence of this iteration is es-sentially independent of h . Hint: use the fact that the eigenvalues of A are jk =-4+2cos jh +2cos kh, j,k = 1 ,...,N, h = 1 N + 1 . (b) How does inuence convergence? 2...
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Linear Algebra Sample Quals - , T = 4-1-1 . . . . . . . . ....

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