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SU Matrix Eigenvectors and Stanford Seal Exam Practice

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Copyright c 2021 Stanford University Department of Mathematics. All rights reserved. Math 51, Spring 2021 Practice problems for Quiz 8 Page 1 of 4 1. (2 points) Suppose the matrix A satisfies A = ST where √ √   √ 1/√3 1/ 2 1/ √6 S = 1/√3 0√ −2/√ 6 , 1/ 3 −1/ 2 1/ 6   0 3 2 T = 0 3 2 0 0 5 Observe that the columns of S are orthonormal. What are the dimensions of N (A) and C(A) (i.e., the null space and column space of A, respectively)? (i) 0 (ii) 1 (iii) 2 (iv) 3 2. (4 points) Suppose A is a 2 × 2 matrix with eigenvalues µ1 , µ2 ; and that B is a symmetric 2 × 2 matrix with eigenvalues λ1 , λ2 . Consider also the following picture of the Stanford seal with two vectors (one black/vertical; one blue/non-vertical) superimposed, and suppose the additional three statements about these vectors, written alongside the figure: 3 • The blue (non-vertical) vector shown above is an eigenvector of A with eigenvalue µ1 . 2 • The black (vertical) vector shown above is an eigenvector of A with eigenvalue µ2 . 1 • The blue (non-vertical) vector shown above is an eigenvector of B with eigenvalue λ1 . 0 -1 -2 -3 -3 -2 -1 0 ...
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