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Linear Algebra with Applications 7th edition Steven J. Leon - Solutions

Let A be a symmetric positive definite n × n matrix. Show that A can be factored into a product QQT, where Q is an n × n matrix whose columns are mutually orthogonal.
Reorder the eigenvalues in Example 2 so that λ1 = 4 and λ2 = 2 and rework the example. In what quadrants will the positive x′ and y′ axes lie? Sketch the graph and compare it to Figure 6.6.3.
In each of the following, find a suitable change of coordinates (i.e., a rotation and/or a translation) so that the resulting conic section is in standard form; identify the curve and sketch the graph. (a) 3x2 + 8xy + 3y2 + 28 = 0 (b) -3x2 + 6xy + 5y2 - 24 = 0
Let A be a symmetric 2 × 2 matrix and let α be a nonzero scalar for which the equation xTAx = α is consistent. Show that the corresponding conic section will be nondegenerate if and only if A is nonsingular.
Which of the following matrices are positive definite? Negative definite? Indefinite?(a)(b)
For each of the following functions, determine whether the given stationary point corresponds to a local minimum, local maximum, or saddle point. (a) f(x, y) = y/x2 + x/y2 + xy (1, 1) (b) f(x, y, z) = x3 + xyz + y2 - 3x (1, 0, 0)
Show that if A is symmetric positive definite then det(A) > 0. Give an example of a 2 × 2 matrix with positive determinant that is not positive definite.
Show that if A is a symmetric positive definite matrix then A is nonsingular and A-1 is also positive definite.
Let A be a symmetric n × n matrix. Show that eA is symmetric and positive definite.
Letand (a) Show that A is positive definite and that xTAx = xTBx for all x ˆˆ R2. (b) Show that B is positive definite, but B2 is not positive definite.
Let A be an n × n symmetric negative definite matrix. (a) What will the sign of det(A) be if n is even? If n is odd? (b) Show that the leading principal submatrices of A are negative definite. (c) Show that the determinants of the leading principal submatrices of A alternate in sign.
Let A be a symmetric positive definite n Ã— n matrix.(a) If kwhere yk ˆˆ Rk and Î²k is a scalar, show that Lk+1 is of the form and determine xk and Î±k in terms of Lk, yk, and Î²k. (b) The leading principal submatrix A1 has Cholesky
Let(a) Compute the LU factorization of A. (b) Explain why A must be positive definite.
Let A be an n × n symmetric positive definite matrix. For each x, y ∈ Rn, define (x, y) = xTAy Show that ( , ) defines an inner product on Rn.
Let A be a nonsingular n × n matrix, and suppose that A = L1D1U1 = L2D2U2, where L1 and L2 are lower triangular, D1 and D2 are diagonal, U1 and U2 are upper triangular, and L1, L2, U1, U2 all have 1's along the diagonal. Show that L1 = L2, D1 = D2, and U1 = U2.
Let A be a symmetric positive definite matrix and let Q be an orthogonal diagonalizing matrix. Use the factorization A = QDQT to find a nonsingular matrix B such that BTB = A.
Let B be an m × n matrix of rank n. Show that BTB is positive definite.
Prove that a 2 × 2 matrix A is reducible if and only if a12a21 = 0.
Prove the Frobenius theorem in the case where A is a 2 × 2 matrix.
We can show that, for an n × n stochastic matrix, λ1 = l is an eigenvalue and the remaining eigenvalues must satisfy |λj| ≤ 1 j = 2,..., n Show that if A is an n × n stochastic matrix with the property that Ak is a positive matrix for some positive integer k, then |λj| < 1 j = 2,..., n
Let A be an n × n positive stochastic matrix with dominant eigenvalue λ1 = 1 and linearly independent eigenvectors x1, x2,..., xn, and let y0 be an initial probability vector for a Markov chain y0, y1 = Ay0, y2 = A1, ... (a) Show that λ1 = 1 has a positive eigenvector x1. (b) Show that ||yj||1 =
Would the results of parts (c) and (d) in Exercise 13 be valid if the stochastic matrix A was not a positive matrix? Answer this same question in the case when A is a nonnegative stochastic matrix and, for some positive integer k, the matrix Ak is positive. Explain your answers. Exercise 13 (c)
Which of the following matrices are reducible? For each reducible matrix, find a permutation matrix P such that PAPT is of the formwhere B and C are square matrices. (a) (b)
Let A be a nonnegative irreducible 3 × 3 matrix whose eigenvalues satisfy λ1 = 2 = |λ2| = |λ3| = Determine λ2 and λ3.
Letwhere B and C are square matrices. (a) If Î» is an eigenvalue of B with eigenvector x = (x1,..., xk)T, show that Î» is also an eigenvalue of A with eigenvector = (x1,..., xk, 0,..., 0)T. (b) If B and C are positive matrices, show that A has a positive real eigenvalue r
If A is an n × n matrix whose eigenvalues are all nonzero, then A is nonsingular. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If A is an n × n matrix, then A and AT have the same eigenvectors. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If A is a 4 × 4 matrix of rank 3 and λ = 0 is an eigenvalue of multiplicity 3, then A is diagonalizable. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If A is a 4 × 4 matrix of rank 1 and λ = 0 is an eigenvalue of multiplicity 3, then A is defective. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
The rank of an m × n matrix A is equal to the number of nonzero singular values of A, where singular values are counted according to multiplicity. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not
If A is symmetric positive definite, then A is nonsingular and A-1 is also symmetric positive definite. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
Let(a) Find the eigenvalues of A. (b) For each eigenvalue, find a basis for the corresponding eigenspace. (c) Factor A into a product XDX-1 where D is a diagonal matrix and then use the factorization to compute A7.
Let A be a 5 × 5 nonsymmetric matrix with rank equal to 3, let B = ATA, and let C = eB. (a) What, if anything, can you conclude about the nature of the eigenvalues of B? Explain. What words best describe the type of matrix that B is? (b) What, if anything, can you conclude about the nature of the
Let A and B be n × n matrices. (a) If A is real and nonsymmetric with Schur decomposition UTUH, then what types of matrices are U and T? How are the eigenvalues of A related to U and T? Explain your answers. (b) If B is Hermitian with Schur decomposition WSWH, then what types of matrices are W and
Let A be a matrix whose singular value decomposition is given byMake use of the singular value decomposition to do each of the following. (a) Determine the rank of A. (b) Find an orthonormal basis for R(A). (c) Find an orthonormal basis for N(A). (d) Find the rank 1 matrix B that is the closest
Let A be a 4 × 4 matrix with real entries that has all 1's on the main diagonal (i.e., a11 = a22 = a33 = a44 = 1). If A is singular and λ1 = 3 + 2i is an eigenvalue of A, then what, if anything, is it possible to conclude about the values of the remaining eigenvalues λ2, λ3, and λ4? Explain.
Let A be a nonsingular n × n matrix and let λ be an eigenvalue of A. (a) Show that λ ≠ 0. (b) Show that 1/λ is an eigenvalue of A-1.
Show that if A is a matrix of the formthen A must be defective.
Given(a) Without computing the eigenvalues of A, show that A is positive definite. (b) Factor A into a product LDLT where L is unit lower triangular and D is diagonal. (c) Compute the Cholesky factorization of A.
Given f(x, y) = x3y + x2 + y2 - 2x - y + 4 has a stationary point (1, 0). Compute the Hessian of f at (1, 0) and use it to determine whether the stationary point is a local maximum, local minimum or saddle point.
GivenY€²(t) = AY(t) Y(0) = Y0wherecompute etA and use it to solve the initial value problem.
Let A be a 4 × 4 real symmetric matrix with eigenvalues λ1 = 1, λ2 = λ3 = λ4 = 0 (a) Explain why the multiple eigenvalue λ = 0 must have three linearly independent eigenvectors x2, x3, x4. (b) Let x1 be an eigenvector belonging to λ1. How is x1 related to x2, x3, x4? Explain. (c) Explain
Let {u1, u2} be an orthonormal basis for C2 and suppose that a vector z can be written as a linear combination z = (5 - 7i)u1 + c2u2 (a) What are the values of uH1z and zHu1? If zHu2 = 1 + 5i, determine the value of c2. (b) Use the results from part (a) to determine the value of ||z||2.
The top matrix on the menu is the diagonal matrixInitially, when you select this matrix, the vectors x and Ax should both be aligned along the positive x axis. What information about an eigenvalue-eigenvector pair is apparent from the initial figure positions? Explain. Rotate x counterclockwise
Set A = ones(10) + eye(10) (a) What is the rank of A - I? Why must λ = 1 be an eigenvalue of multiplicity 9? Compute the trace of A using the MATLAB function trace. The remaining eigenvalue λ10 must equal 11. Why? Explain. Compute the eigenvalues of A by setting e = eig(A). Examine the
Consider the matricesNote that the two matrices are the same except for their (2, 2) entries. (a) Use MATLAB to compute the eigenvalues of A and B. Do they have the same type of eigenvalues? The eigenvalues of the matrices are the roots of their characteristic polynomials. Use the following MATLAB
Set C = triu(ones(4), 1) + diag([l, -1], -2) [X, D] = eig(C) Compute X-1CX and compare the result to D. Is C diagonalizable? Compute the rank of X and the condition number of X. If the condition number of X is large, the computed values for the eigenvalues may not be accurate. Compute the reduced
Construct a defective matrix by setting A = ones(6); A = A - tril(A) - triu(A, 2) It is easily seen that λ = 0 is the only eigenvalue of A and that its eigenspace is spanned by e1. Verify this by using MATLAB to compute the eigenvalues and eigenvectors of A. Examine the eigenvectors using format
Generate a matrix A by setting B = [-1, -1; 1, 1], A = [zeros(2), eye(2); eye(2), B] (a) The matrix A should have eigenvalues λ1 = 1 and λ2 = -1. Use MATLAB to verify this by computing the reduced row echelon forms of A - I and A + I. What are the dimensions of the eigenspaces of λ1 and λ2? (b)
Suppose that 10,000 men and 10,000 women settle on an island in the Pacific that has been opened to development. Suppose also that a medical study of the settlers finds that 200 of the men are color-blind and only 9 of the women are color-blind. Let x(1) denote the proportion of genes for color
Construct a complex Hermitian matrix by setting j = sqrt(-1); A = rand(5) + j * rand(5); A = (A + A')/2 (a) The eigenvalues of A should be real. Why? Compute the eigenvalues and examine your results using format long. Are the computed eigenvalues real? Compute also the eigenvectors by setting [X,
The third matrix on the menu is just the identity matrix I. How do x and Ix compare geometrically as you rotate x around the unit circle? What can you conclude about the eigenvalues and eigenvectors in this case?
Let A be a nonsingular 2 × 2 matrix with singular value decomposition A = USVT and singular values s1 = s11, .s2 = s22. Explain why each of the following are true. (a) AV = US. (b) Av1 = s1u1 and Av2 = s2u2. (c) v1 and v2 are orthogonal unit vectors and the images Av1 and Av2 are also
Use the command eigshow(A) to apply the eigshow utility to the matrix A. Click on the eig/(svd) button to switch into the svdshow mode. The display in the figure window should show a pair of orthogonal vectors x, y and their images Ax and Ay. Initially, the images of x and y should not be
Use the following MATLAB commands to construct a symbolic function. syms x y f = (y + l)^3 + x * y^2 + y^2 - 4 * x * y - 4 * y + l Compute the first partials of f and the Hessian of f by setting fx = dif f (f, x), fy = dif f (f, y) H = [diff (fx, x), diff (fx, y); diff (fy, x), diff (fy, y)] We can
Set C = ones(6) + 7 * eye(6) and [X, D] = eig(C) (a) Even though λ = 7 is an eigenvalue of multiplicity 5, the matrix C cannot be defective. Why? Explain. Check that C is not defective by computing the rank of X. Compute also XTX. What type of matrix is X? Explain. Compute also the rank of C - 7I.
For various values of k, form an k × k matrix A by setting A = 2 * eye(k) - diag(ones(k - 1, 1), 1) - diag(ones(k - 1, 1), -1) In each case, compute the LU factorization of A and the determinant of A. If A is an n × n matrix of this form, what will its LU factorization be? What will its
For any positive integer n, the MATLAB command P = pascal(n) will generate an n Ã— n matrix P whose entries are given byThe name pascal refers to Pascal's triangle, a triangular array of numbers that is used to generate binomial coefficients. The entries of the matrix P form a section
The fourth matrix has 0's on the diagonal and l's in the off-diagonal positions. Rotate the vector x around the unit circle and note when x and Ax are parallel. Based on these observations, determine the eigenvalues and the corresponding unit eigenvectors. Check your answers by multiplying the
Investigate the next three matrices on the menu. You should note that for the last two of these matrices the two eigenvalues are equal. For each matrix, how are the eigenvectors related? Use MATLAB to compute the eigenvalues and eigenvectors of these matrices.
The last item on the eigshow menu will generate a random 2 × 2 matrix each time that it is invoked. Try using the random matrix 10 times and in each case determine whether the eigenvalues are real. What percentage of the 10 random matrices had real eigenvalues? What is the likelihood that two real
Let A be an n × n matrix with distinct real eigenvalues λ1,λ2 . . . . λn. Let λ be a scalar that is not an eigenvalue of A and let B = (A - λI)-1. Show that (a) The scalars μj = l/(λj - λ), j = 1,... ,n are the eigenvalues of B. (b) If xJ is an eigenvector of B belonging to μj, then xj
Let x = (x1. . . .xn)T be an eigenvector of A belonging to Î». If |xi| = ||x||ˆž, show that(a)(b)
Let A be a matrix with eigenvalues Î»1. . . . . . Î»n and let Î» be an eigenvalue of A + E.Let X be a matrix that diagonalizes A and let C = X-1EX. Prove(a) For some i(b)
Let Ak = QkRk. k= 1,2,... be the sequence of matrices derived from A = A1 by applying the QR algorithm. For each positive integer k, define Pk = Q1Q2 . . . .Qk and Uk = Rk . . . .R2R1 Show that PkAk+1 = APk For all k ≥ 1.
Let A be an m × n matrix with singular value decomposition U∑VT and suppose that A has rank r, where r < n. Let b ∈ Rm. Show that a vector x ∈ Rn minimizes ||b - Ax||2 if and only if x = A+b + cr+1vr+1 + . . . + cnvn where cr+1, . . . ., cn are scalars
Show each of the following:(a) (A+)+ = A(b) (AA+)2 = AA+(c) (A+A)2 = A+A
Let A = XYT, where X is an m × r matrix, YT is an r × n matrix, and XTX and YTY are both nonsingular. Show that the matrix B = Y (YTY)-1(XTX)-1 XT satisfies the Penrose conditions and hence must equal A+. Thus A+ can be determined from any factorization of this form.
Given(a) Use Householder transformation to reduce A to the formand apply the same transformations to be.
Show that the pseudoinverse A+ satisfies the four Penrose conditions.
Let B be any matrix that satisfies Penrose conditions 1 to 3 and let x = Bb. Show that x is a solution to the normal equations ATAx = ATb.
If x ˆˆ Rm, we can think of x as an m Ã— 1 matrix. If x is nonzero we then can define a 1 Ã— m matrix X byShow that X and x satisfy the four Penrose conditions and consequently that
Show that if A is a m × n matrix of rank n then A+ = (ATA)-1AT
Let S be an m × n matrix and let b ∈ Rm. Show that b ∈ R(A) if and only ifb =AA+b
If a, b, and c are floating - point numbers, then fl(fl(a + b) + c) = fl(a + fl(b + c)) In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If two m × n matrices A and B are close in the sense that ||A - B||2 < ∈ for some small positive number6, then their pseudoinverses will also be close; that is, ||A+ - B+||2 < δ, for some small positive number δ. In the case of a true statement, explain or prove your answer. In the case of a
The computation of A(BC) requires the same number of floating-point operations as the computation of (AB)C In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If A is a nonsingular matrix and a numerically stable algorithm is used to compute the solution to a system Ax = b, then the relative error in the computed solution will always be small. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example
If A is a symmetric matrix and a numerically stable algorithm is used to compute the eigenvalues of Ax = b, then the relative error in the computed eigenvalues should always be small. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to
If A is a nonsymmetric matrix and a numerically stable algorithm is used to compute the eigenvalues of Ax = b, then the relative error in the computed eigenvalues should always be small. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example
If both A-1 and the LU factorization of an n × n matrix A have already been computed, then it is more efficient to solve a system Ax = b by multiplying A-1b rather than solving LUx = In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to
If A is a symmetric matrix, ||A||1 = ||A||∞. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If A is an m × n matrix, ||A||2 = ||A||F. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.
If the coefficient matrix A in a least squares problem has dimensions m × n and rank n, then the three methods of solution discussed in Section 7, the normal equations, the QR factorization, and the singular value decomposition, will all compute highly accurate solutions. In the case of a true
Let A and B be n x n matrices and let x be a vector in Rn. How many scalar additions and multiplications are required to compute (AB)x and how many are necessary to compute A(Bx)? Which computation is more efficient?
LetUse Householder matrices to transform A into a 4 Ã— 2 upper triangular matrix R. Apply the same Householder transformations to b and then compute the least squares solution to the system Ax = b.
Given(a) Use Gaussian elimination with partial pivoting to solve Ax = b. (b) Write the permutation matrix P that corresponds to the pivoting strategy in part (a) and determine the LU factorization of PA. (c) Use P, L, and U to solve the system Ax = c.
Show that if Q is any 4 × 4 orthogonal matrix then ||Q||2 = 1 and ||Q||F = 2.
Given(a) Determine the values of ||H||1 and ||H-1||1. (b) When the system Hx = b is solved using MATLAB and the computed solution xÊ¹ is used to compute a residual vector r = b - HxÊ¹, it turns out that ||r||1= 0.36 Ã—10-11. Use this information to determine a
Let A be a 10 × 10 matrix with condition number 5 x 106. Suppose that the solution to a system Ax = b is computed using 15-digit decimal arithmetic and the relative residual ||r||∞/||b||∞, turns out to be approximately twice the machine epsilon. How many digits of accuracy would you expect to
Let x = (1,2, -2)T. (a) Find a Householder matrix H such that Hx is a vector of the form (r, 0, 0)T. (b) Find a Givens transformation G such that Gx is a vector of the form (1, s, 0)T.
Let Q be an n × n orthogonal matrix and let R be an n × n upper triangular matrix. If A = QR and B = RQ, how are the eigenvalues and eigenvectors of A and B related? Explain.
LetEstimate the largest eigenvalue of A and a corresponding eigenvector by doing five iterations of the power method. You may start with any nonzero vector x0.
LetThe singular value decomposition of A is given by Use the singular value decomposition to find the least squares solution to the system Ax = b that has the smallest 2 n-orma.
To plot y = sin(x), we must define vectors of x and y values and then use the plot command. This can be done as follows: x = 0:0.1:6.3; y = sin(x); plot(x, y) (a) Let us define a rotation matrix and use it to rotate the graph of y = sin(x). Set t=pi/4; c = cos(t): s = sin(t); R
GivenEnter the matrix A in MATLAB and compute its singular values by setting s = svd(A). (a) To obtain the full singular value decomposition of A, set [U, D, V]= svd(A) Compute the closest matrix of rank 1 to A by setting B = s(1) * U(:, 1) * V(:, 1)Ê¹ How are the row vectors of B
Set A = round(10 * rand(10, 5)) and s = svd(A) (a) Set [U, D, V] = svd(A); D(5,5)=0; B = U*D*Vʹ The matrix B should be the closest matrix of rank 4 to A (where distance is measured in terms of the Frobenius norm). Compute ||A||2 and ||6||2. How do these values compare? Compute and
Set A = rand(8, 4) * rand(4, 6), [U, D, V]= svd(A) (a) What is the rank of A? Use the column vectors of V to generate two matrices V l and V2 whose columns form orthonormal bases for R(AT) and N(A), respectively. Set p = V2 * V2ʹ, r = P * rand(6, 1), w = Aʹ * rand(8, 1) If r and w had been
With each A ˆˆ RnÃ—n we can associate n closed circular disks in the complex plan. The ith disk is centered at aij and has radiusEach eigenvalue of A is contained in at least one of the disks. If k of the Gerschgorin disks form a connected domain in the complex plane that is
Construct a matrix C as follows. Set A = round(100 * rand(4)) L = tril(A,-1) + eye(4) C = L*Lʹ The matrix C is a nice matrix in that it is a symmetric matrix with integer entries and its determinant is equal to 1. Use MATLAB to verify these claims. Why do we know ahead of time that the determinant

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