Math 1b Practical — Problem Set 8— Revised March 3, 2010Due 3:00pm, Monday, March 8, 20101. (i) Sketch the digraph corresponding to the following matrix of 0’s and 1’s.A =⎛⎜⎝1100001000011100⎞⎟⎠.Label the four vertices 1, 2, 3, 4.(ii) There are two directed walks from vertex 1 to vertex 4 in the digraph that havelength 6, and these (described by the sequence of seven vertices that are visited) are1234234 and 1111234.List all directed walks from 2 to 4 of length 6 in this digraph. List all directed walks from1 to 2 of length 6 in this digraph.(iii) Compute A6(use a calculator or computer) to check that you got the correctnumber of walks. What is the smallest value of m for which Am>O(you may again usea calculator or a computer)? [The number of directed walks of length k from a vertex i toavertexj in the digraph corresponding to a matrix A of 0’s and 1’s is the (i, j)-entry ofAk.][Remark: A is not a probability matrix. But if we divide columns 2 and 4 by 2 to get amatrix P ,thenP is a probability matrix, and it has the same digraph as A.]2. LetA =11 88 −1.An orthogonal matrix is a square matrix U so that UU = UU= I. Find an orthogonalmatrix U so that UAU is a diagonal matrix with the eigenvalues of A on the diagonal.3. LetS =⎛⎜⎜⎜⎜⎜⎝511111151111115111111511111151111115⎞⎟⎟⎟⎟⎟⎠.Find an orthonormal basis for RR6consisting of eigenvectors of S. [First find an or-thogonal basis; then normalize. There is no need to use Gram-Schmidt on the eigenvectorscorresponding to a given eigenvalue λ if you can find orthogonal eigenvectors by guessingand checking.]4. (i) Let e be an eigenvector of a real symmetric matrix S and let U = {e}⊥.Provethatif x ∈ U,thenSx ∈ U.(ii) Show that if U is a orthogonal matrix, then the corresponding linear transforma-tion from RRnto RRnpreserves angle. That is, the angle between x and y is the same asthat between Ux and Uy. [You will have to look up the definition of angle somewhere.Then this should be easy.]5. Find a positive definite symmetric matrix A so thatA2=5 −3−32using the method described on page 3 of the Notes on Isometries.[I suggest you use a matrix calculator or computer, because this is surprisingly messyeven for a 2 × 2 matrix. Do it numerically or symbolically; take your choice. Thereis a simple square root with integer entries, but there are many square roots, and yourcalculations may give a different answer.][Mathematica: You can use Aˆ.5 to take the square root of every entry of the matrix A.This is the entry-wise square root, not a square root with respect to matrix
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