Math 2B Spring 13 Chapters 5 and 6 Test
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1 Math B Spring 3 Chapters 5 and 6 Test Name Give complete solutions to all problems. You can use technology to check your results, but be sure to detail how they are derived. Do not copy other people s work Find the LU factorization of and use it to compute the determinant, in terms of n Assuming we do not perform any row permutations, what is the 5 th and last pivot when transforming the following matrix into an upper triangular matrix by row eliminations? Explain Is there an 5 3 factorization of. Explain Find the Cholesky factorization of. Is positive definite? Explain The implicit diagonalization Λ occurs in eigenvalue problems, where Λ is the diagonal matrix of eigenvalues. a. Describe the significance of the individual columns of in relation to the diagonal entries in Λ. b. Describe two uses of the diagonalization factorization Let a. Find the eigenvalues of. b. Find an orthogonal matrix such that is diagonal. c. Find an expression for exp. 7. (a) Argue that the eigenvalues of a permutation matrix must all have a modulus equal to (b) What are the eigenvalues of ? (c) Express the eigenvalues in polar form If 0 3/ 0 and is invertible. 0 0 /6 a. What is the determinant of?
2 b. What are the eigenvalues of? Of? And of? c. Find a factored form of in terms of S and the diagonal matrix above. d. Calculate the trace of and then the trace of. 9. The quadratic 5 85 describes a tilted ellipse in the xy-plane. a. Use eigenvalue methods to find the lengths of the major and minor axes. b. Find the directions of the axes of this tilted ellipse in the xy-plane 0. One important use of SVD is in finding orthonormal bases for some or all of the four fundamental subspaces. Other uses include finding the rank of a matrix and solving least squares problems. You can extract subspace information by selecting appropriate columns from or. The code below shows Octave output for a specific magic matrix. Use the SVD output to find an orthonormal basis for the column space and the nullspace. What is the rank of? A=magic(4) % matrix in question A = octave:> help svd -- Loadable Function: S = svd (A) -- Loadable Function: [U, S, V] = svd (A) -- Loadable Function: [U, S, V] = svd (A, ECON) Compute the singular value decomposition of A A = U*S*V' The function 'svd' normally returns only the vector of singular values. When called with three return values, it computes U, S, and V. For example, svd (hilb (3)) returns ans = and [u, s, v] = svd (hilb (3)) returns u = s = v = If given a second argument, `svd' returns an economy-sized decomposition, eliminating the unnecessary rows or columns of U or V.
3 . Suppose that is 5 by 3 and Σ is its SVD, where Σ a. Find the eigenvalues of using Σ. b. Remember properties of orthogonal matrices and evaluate the determinants, and.. Write in terms of a symmetric matrix and vector,,,. What is the rank of?
4 Math B Spring 3 Chapters 5 and 6 Test Solutions Find the LU factorization of and use it to compute the determinant, in terms of n SOLN: The determinant of the product is the product of determinants. The determinant of L is, while the determinant of U is 5. Thus 5 and 5.. Assuming we do not perform any row permutations, what is the 5 th and last pivot when transforming the following matrix into an upper triangular matrix by row eliminations? Explain SOLN: The fifth pivot is given by,,, Is there an 5 3 factorization of. Explain SOLN: No, A is not symmetric Find the Cholesky factorization of. Is positive definite? Explain SOLN: / / / / / / / 0 0 / / /3 0 0 / / / / /
5 0 0 0 The Cholesky factorization is then / 6/ 0 0 where 0 6/3 3/ / 5/ or, as Mathematica has it,. Yes, is positive definite since the pivots have the same signs as the eigenvalues and the pivots are all positive. 5. The implicit diagonalization Λ occurs in eigenvalue problems, where Λ is the diagonal matrix of eigenvalues. a. Describe the significance of the individual columns of in relation to the diagonal entries in Λ. SOLN: The nth column of S is the eigenvector for the eigenvalue at Λ,. b. Describe two uses of the diagonalization factorization. SOLN: () It s much easier to compute powers of a matrix if it s in diagonalized form. () It s easier to compute the determinant. (3) Get answer 5b right on the exam Let a. Find the eigenvalues of Factoring, we find 7 0 the eigenvalues are and the repeated 7. b. Find an orthogonal matrix such that is diagonal The eigenvector for spans the null space of 8 ~ / Now for 7 the situation is different. 7 ~0 0 0 span Trouble is, of course, that while these vectors span the space, they re not orthonormal. Gram Schmidt! Take. Then 0 So 0. Ha! They re already perpendicular! 0 4/5 0 /5 Normalizing, we get the matrix 0 0 /3 5/5 4 5/5 /3 5/5 5/5 /3 0 5/3
6 3 4 /3 5/5 4 5/5 0 0 /3 /3 /3 Thus 6 /3 5/5 5/ /5 5/ /3 0 5/ /5 5/5 5/3 and that s a fact. c. Find an expression for exp. SOLN: /3 5/5 4 5/5 0 0! /3 /3 /3 exp /3 5/5 5/5! 0 0! 5/5 5/5 0 /3 0 5/ /5 5/5 5/3! /3 5/5 4 5/5 0 0 /3 /3 /3 /3 5/5 5/ /5 5/5 0 /3 0 5/ /5 5/5 5/3 7. (a) Argue that the eigenvalues of a permutation matrix must all have a modulus equal to. SOLN: Assume that is an eigenvalue of a permutation matrix and is the corresponding eigenvector so that. Both and may be complex (and often are). Now look at the sum of the moduli of the 's, i.e.. As is a permutation matrix, has the same entries as except that they are in a different order. Thus That sure makes it seem as though the eigenvalues have modulus. To gather more evidence in support of this, note the diagonalization of 0 / / 0 / / 0 / / which is nicely interpreted as first rotate clockwise by 0 / / 45, then reflect across the x-axis, then rotate counterclockwise by 45. But the values are all real. How about ? Here the characteristic equation is 0 so the eigenvalues are cube roots of unity: 0 0,. For, the eigenvector is. For the eigenvector is and for the eigenvector is Thus Note that a non symmetric matrix has been factored as a product of symmetric matrices. These are the strange doings of complex valued matrices Let s push on to 44 before we go to our 55 problem.
7 has the characteristic equation 0 whose roots are,. I m not going to type out all the details here, but it s easy to verify these (b) What are the eigenvalues of ? SOLN: The characteristic equation is This can be factored by grouping, so 0 so the eigenvalues are with as a repeated eigenvalue. Mathematica writes the roots this way:,,,, which seems a tad peculiar. If you ask for the Eigensystem[] it reports the expected eigenvalues: 3 3, and corresponding eigenvectors: ,,0,0,0 0,0,,,,,0,0,0 0,0,,, 0,0,,, These shouldn t be too hard to verify. For, ~ So the null space is spanned by,,0,0, For, ~ whose null space is spanned by , (note that rows and 5, as there are no pivots in those columns, comprise while the negation of the elements of those non-pivot columns fill in the rest.) As for the complex conjugate eigenvalues, well, for
8 , whose null space is spanned by 0,0, ~ ,, and for conjugate eigenvalue, , whose null space is spanned by 0,0,, ~ ,. Thus the diagonalization is almost as shown below apparently there are some problems with scaling (the upper left block has a different scale than the lower right block) and again the permutation thing somehow the 3 rd row remains to be pushed to the bottom (c) Express the eigenvalues in polar form. SOLN: cos sin exp If 0 3/ 0 and is invertible. 0 0 /6 a. What is the determinant of? SOLN: 0 3/ 0 0 3/ 0 0 3/ /6 0 0 /6 0 0 /6 b. What are the eigenvalues of? Of? And of? SOLN: Λ Λ Λ so the eigenvalue matrix for is / 0 and the eigenvalues of are, ½ and 5/ /6 Similarly, Λ ΛΛ ΛΛ so the eigenvalues of are 3/, 3/6 and 37/6. The eigenvalues of are ½, /3 and 6. a. Find a factored form of in terms of S and the diagonal matrix above.
9 0 0! Λ 0 0!! 0 3/ 0! /6! b. Calculate the trace of and then the trace of. SOLN: The trace of is the sum of its eigenvalues: /3. The trace of is then / / 9. The quadratic 5 85 describes a tilted ellipse in the xy-plane. a. Use eigenvalue methods to find the lengths of the major and minor axes. SOLN: 5 85, The characteristic polynomial has eigenvalue 9 with eigenvector (,) and eigenvalue with eigenvector (, ) so the diagonalization is /4 sin /4 cos sin /4 cos / cos /4 sin /4 Thus the ellipse is sin /4 cos /4 congruent to 9 whose major axis has length and the minor axis has length /3. b. Find the directions of the axes of this tilted ellipse in the xy-plane SOLN: The first rotation is clockwise, so the major axis is pointed in the direction of ContourPlot5 85,,,,,,, Axes True, Frame False 0. One important use of SVD is in finding orthonormal bases for some or all of the four fundamental subspaces. Other uses include finding the rank of a matrix and solving least squares problems. You can extract subspace information by selecting appropriate columns from or. The code below shows Octave output for a specific magic matrix. Use the SVD output to find an orthonormal basis for the column space and the nullspace. What is the rank of? SOLN: A=magic(4) % matrix in question A = octave:5> [u, s, v] = svd(a) u = s = Diagonal Matrix e e e e-06
10 v = Clearly that last eigenvalue is machine zero. Well, let s be as honest as we can. The column vectors of are the eigenvectors of The eigenvalues of are values of that make singular. That is, so that The eigenvector for 56 is the unit vector,,, that spans the null space of ~ The eigenvector for 30 is the unit vector,,, that spans the null space of ~ The eigenvector for 34 is the unit vector,,, that spans the null space of ~ so the nullspace is empty. Note that Hmmm Since ~ we might instead use ,,, but that s for the zero eigenvector. I do not fully grasp what s going on here I suspect it s something deep. In any case, let s turn our attention to. The column vectors are the eigenvectors of Since is symmetric, and the singular value decomposition has well
11 Where is pretty much what Octave got is likewise As to, 56 34, but doesn t jibe with what the machine algorithms are getting, which is 5 Go figure. While you re at it, go transfer to university and find yourselves some answer to this!. Suppose that is 5 by 3 and Σ is its SVD, where Σ a. Find the eigenvalues of using Σ SOLN: Σ Σ suggests that the eigenvalues are 6, 4, b. Remember properties of orthogonal matrices and evaluate the determinants, and. SOLN: Clearly will be singular, while will not, so 0 while 64.. Write in terms of a symmetric matrix and vector,,,. What is the rank of? SOLN: Take 0,,, Clearly the rank of A is.
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