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linear algebra

Applied Linear Algebra 1st edition Peter J. Olver, Cheri Shakiban - Solutions

Let L = L*: Rn → Rn be a self-adjoint linear transformation with respect to the inner product (∙.∙). Prove that all its eigenvalues are real and the eigenvectors are orthogonal.
Using the set-up in the proof of Theorem 8.20, explain whyThen prove that bjj = bji, justifying the statement that the restriction of A to VŠ¥ is represented by a symmetric matrix.
The difference map ∆: Cn → Cn is defined as ∆ = S - I, where S is the shift map of Exercise 8.2.13.(a) Write down the matrix corresponding to A.(b) Prove that the sampled exponential vectors w0, ...,wn-1 from (5.90) form an eigenvector basis of ∆. What are the eigenvalues?(c) Prove that K =
An n Ã— n circulant matrix has the formin which the entries of each succeeding row are obtained by moving all the previous row's entries one slot to the right, the last entry moving to the front. (a) Check that the shift matrix S of Exercise 8.2.13, the difference matrix ˆ†,
Write out the spectral factorization of the following matrices:(a)(b) (c) (d)
Write out the spectral factorization of the matrices listed in Exercise 8.4.1.(a)(b) (c) (d) (e)
Construct a symmetric matrix with the following eigenvectors and eigenvalues, or explain why none exists: (a) λ1 = 1, v1 = (3/5, 4/5)T, λ2 = 3, v2 = (-4/5, 3/5)T (b) λ1 = -2, v1 = (1, -l)T, λ2 = l, v2 = (1, l)T (c) λ1 = 3, v1 = (2, -l)T, λ2 = -1, v2 = (-1, 2)T (d) λ1 = 2, v1 = (2, l)T λ2 =
Use the spectral factorization to diagonalize the following quadratic forms: (a) x2 - 3xy + 5y2 (b) 3x2 + 4x y + 6y2 (c) x2 + 8xz + y2 + 6yz + z2 (d) 3/2x2 - xy - xz + y2 + z2 (e) 6x2 - 8xy + 2xz + 6y2 - 2yz + 11z2
Find the eigenvalues and eigenvectors of the matrix(a) Use the eigenvalues to compute the determinant of A. (b) Is A positive definite? Why or why not? (c) Find an orthonormal eigenvector basis of R3 determined by A or explain why none exists. (d) Write out the spectral factorization of A if
Let u1,...,un be an orthonormal basis of Rn. Prove that it forms an eigenvector basis for some symmetric n × n matrix A. Can you characterize all such matrices?
Determine whether the following symmetric matrices are positive definite by computing their eigenvalues. Validate your conclusions by using the methods from Chapter 4.(a)(b) (c) (d)
True or false: A matrix with a real orthonormal eigenvector basis is symmetric.
Prove that any quadratic form can be written aswhere A, are the eigenvalues of A and Î¸i = Š€(x, vi) denotes the angle between x and the ith eigenvector.
An elastic body has stress tensorFind the principal stretches and principal directions of stretch.
Let K be a positive definite 2 × 2 matrix. (a) Explain why the quadratic equation xT Kx = 1 defines an ellipse. Prove that its principal axes are the eigenvectors of K, and the semi-axes are the reciprocals of the square roots of the eigenvalues. (b) Graph and describe the following curves: (i) x2
Let K be a positive definite 3 × 3 matrix.(a) Prove that the quadratic equation xTKx = 1 defines an ellipsoid in R3. What are its principal axes and semi-axes?(b) Describe the surface defined by the quadratic equation 11x2 - 8xy + 20y2 - 10xz + 8yz + 11z2 = 1.
Prove that A = AT has a repeated eigenvalue if and only if it commutes, AJ = JA, with a nonzero skew-symmetric matrix: JT = - J ≠ 0.
(a) Prove that every positive definite matrix K has a unique positive definite square root, i.e., a matrix B > 0 satisfying B2 = K.(b) Find the positive definite square roots of the following matrices:
Find all positive definite orthogonal matrices.
The Polar Decomposition: Prove that every invertible matrix A has a polar decomposition, written A = QB, into the product of an orthogonal matrix Q and a positive definite matrix B > 0. Show that if det A > 0, then Q is a proper orthogonal matrix.
Prove that a symmetric matrix is negative definite if and only if all its eigenvalues are negative.
Find the polar decompositions A = QB, as defined in Exercise 8.4.29, of the following matrices:(a)(b) (c) (d) (e)
The Spectral Decomposition:(i) Let A be a symmetric matrix with eigenvalues Î»1,... ,Î»n and corresponding orthonormal eigenvectors Î¼1, ...,Î¼1. Let Pk = uk uTk be the orthogonal projection matrix onto the eigenline spanned by uk, as defined in
The Spectral Theorem for Hermitian Matrices. Prove that any complex Hermitian matrix can be factored as H = U Λ U† where U is a unitary matrix and Λ is a real diagonal matrix.
Find the spectral factorization, as in Exercise 8.4.32, of the following Hermitian matrices:(a)(b) (c)
Write down and solve an optimization principle characterizing the largest and smallest eigenvalue of the following positive definite matrices:
Write down a maximization principle that characterizes the middle eigenvalue of the matrices in parts (c-d) of Exercise 8.4.36.Exercise 8.4.36(c)(d)
What are the minimum and maximum values of the following rational functions: (a) 3x2 - 2y2 / x2 + y2 (b) x2 - 3xy + y2 / x2 + y2 (c) 3x2 + xy + 5y2 / x2 + y2 (d) 2x2 + xy + 3xz + 2y2 + 2z2 / x2 + y2 + z2
How many orthononnal eigenvector bases does a symmetric n × n matrix have?
Let K > 0. Prove the product formula max {xT K x | ||x|| = 1} ∙ min {xT K-1x | ||x|| = 1 } =1.
Write out the details in the proof of Theorem 8.30. Theorem 8.30 Let A be a symmetric matrix with eigenvalues λ1 ≥ λ2 ≥ ... ≥ λn and corresponding orthogonal eigenvectors v1,..., vn. Then the maximal value of the quadratic form q(x) = xT A x over all unit vectors that are orthogonal to the
Reformulate Theorem 8.30 as a minimum principle for intermediate eigenvalues.
Under the set-up of Theorem 8.30, explain why
(a) Let K . M be positive definite n Ã— n matrices and Î»1 ‰¥ ... ‰¥ Î»n be their generalized eigenvalues, as in Exercise 8.4.9. Prove that that the largest generalized eigenvalue can be characterized by the maximum principleÎ»1 = max {xTKx | xTMx = 11}.(b) Prove the alternative
Use Exercise 8.4.45 to find the minimum and maximum of the following rational functions: (a) 3x2 + 2y2 / 4x2 + 5y2 (b) x2 - x y + 2y2 / 2x2 - xy + y2 (c) 2x2 + 3y2 + z2 / x2 + 3y2 + 2z2 (d) 2x2 + 6xy + 11y2 + 6yz + 2z / x2 + 2xy + 3y2 + 2yz + z2
Let(i). Write down necessary and sufficient conditions on the entries a, b, c, d that ensures that A has only real eigenvalues.(ii).Verify that all symmetric 2 Ã— 2 matrices satisfy your conditions.(iii). Note down a non-symmetric matrix that full fill your conditions.
Let(a) Write down necessary and sufficient conditions on the entries a, b, c, d that ensures that A has only real eigenvalues. (b) Verify that all symmetric 2 Ã— 2 matrices satisfy your conditions. (c) Write down a non-symmetric matrix that satisfies your conditions.
Let AT = -A be a real, skew-symmetric n Ã— n matrix.(a) Prove that the only possible real eigenvalue of A is Î» = 0.(b) More generally, prove that all eigenvalues Î» of A are purely imaginary, i.e., Re Î» = 0.(c) Explain why 0 is an eigenvalue of A whenever n is odd.(d) Explain why, if n =
(a) Prove that every eigenvalue of a Hermitian matrix A, satisfying AT = as in Exercise 3.6.49. is real.(b) Show that the eigenvectors corresponding to distinct eigenvalues are orthogonal under the Hermitian dot product on Cn.(c) Find the eigenvalues and eigenvectors of the following Hermitian
Let M > 0 be a fixed positive definite n × n matrix. A nonzero vector v ≠ 0 is called a generalized eigenvector of the n × n matrix K if Kv = λMv, v ≠ 0, (8.31) where the scalar λ is the corresponding generalized eigenvalue. (a) Prove that λ is a generalized eigenvalue of the matrix K if
Compute the generalized eigenvalues and eigenvectors, as in (8.31), for the following matrix pairs. Verify orthogonality of the eigenvectors under the appropriate inner product.
Find the singular values of the following matrices:(a)(b) (c) (d) (e) (f)
True or false: The singular values of AT are the same as the singular values of A.
Prove that if A is square, nonsingular, then the singular values of A-1 are the reciprocals of the singular values of A. How are their condition numbers related?
Let A be a nonsingular square matrix. (a) Prove that the product of the singular values of A equals the absolute value of its determinant: σ1 σ2 ... σn = |det A|. (b) Does their sum equal the absolute value of the trace: σ1 + ... + σn = |tr A|? (c) Show that if det A < 10-k, then its minimal
Let A be a nonsingular 2 Ã— 2 matrix with singular value decomposition A = P ˆ‘ QT and singular values Ïƒ1 ‰¥ Ïƒ2 > 0.(a) Prove that the image of the unit (Euclidean) circle under the linear transformation defined by A is an ellipse. E = {Ax| ||x||
Write out the singular value decomposition (8.40) of the matrices in Exercise 8.5.1.Matrices in Exercise 8.5.1(a)(b) (c) (d) (e) (f)
LetWrite down the equation for the ellipse E = {Ax | ||x|| = 1} and draw a picture. What are its principal axes? Its semi-axes? Its area?
Letand let E = {y = Ax | ||x|| = 1} be the image of the unit Euclidean sphere under the linear map induced by A. (a) Explain why E is an ellipsoid and write down its equation. (b) What are its principal axes and their lengths- the semi-axes of the ellipsoid? (c) What is the volume of the solid
Optimization Principles for Singular Values: Let A be any nonzero m x n matrix. Prove that (a) σ1 = max { ||Au|| | ||u|| = 1}. (b) Is the minimum the smallest singular value? (c) Can you design an optimization principle for the intermediate singular values?
Find the pseudoinverse of the following matrices:(a)(b) (c) (d) (e) (f) (g)
Use the pseudoinverse to find the least squares solution of minimal norm to the following linear systems: (a) x + y = 1 3x + 3y = -2 (b) x + y + z = 5 2x - y + z = 2 (c) x - 3y = 2 2x + y = -1 x + y = 0
Prove that the pseudoinverse satisfies the following identities: (a) (A+)+ = A (b) A A+A = A (c) A+A A+ = A+ (d) (AA+)T = AA+ (e) (A+A)T = A+A
Suppose b ∈ mg A and kerA = {0}. Prove that x* = A+b is the unique solution to the linear system Ax = b. What if ker A ≠ {0}?
(a) Construct the singular value decomposition of the shear matrix(b) Explain how a shear can be realized as a combination of a rotation, a stretch, followed by a second rotation.
Find the condition number of the following matrices. Which would you characterize as ill conditioned?
Solve the following systems of equations using Gaussian Elimination with three-digit rounding arithmetic. Is your answer a reasonable approximation to the exact solution? Compare the accuracy of your answers with the condition number of the coefficient matrix, and discuss the implications of
Compute the singular values and condition numbers of the 2 × 2, 3 × 3. and 4 × 4 Hilbert matrices. What is the smallest Hilbert matrix with condition number larger than 106?
(a) What are the singular values of a 1 × n matrix? (b) Write down its singular value decomposition. (c) Write down its pseudoinverse.
Answer Exercise 8.5.7 for an m × 1 matrix. Exercise 8.5.7 (a) What are the singular values of a 1 × n matrix? (b) Write down its singular value decomposition. (c) Write down its pseudoinverse.
True or false: Every matrix has at least one singular value.
Establish a Schur Decomposition for the following matrices:(a)(b) (c) (d) (e) (f)
Write down a formula for the inverse of a Jordan block matrix.
Suppose you know all eigenvalues of a matrix as well as their algebraic and geometric multiplicities. Can you determine the matrix's Jordan canonical form?
True or false: If w1,... , wj is a Jordan chain for a matrix A, so are the scalar multiples cw1,... , cwj for any c ≠ 0.
True or false: If A has Jordan canonical form J, then A2 has Jordan canonical J2.
(a) Give an example of a matrix A such that A2 has an eigenvector that is not an eigenvector of A. (b) Show that, in general, every eigenvalue of A2 is the square of an eigenvalue of A.
Let A and B be n × n matrices. According to Exercise 8.2.23, the matrix products A B and B A have the same eigenvalues. Do they have the same Jordan form?
Prove Lemma 8.53. Lemma 8.53 The n × n Jordan block matrix Jλ,n has a single eigenvalue, λ, and a single independent eigenvector, e1. The standard basis vectors e1,... , en form a Jordan chain for Jλ,n.
(a) Prove that a Jordan block matrix J0,n with zero diagonal entries is nilpotent. as in Exercise 1.3.13. (b) Prove that a Jordan matrix is nilpotent if and only if all its diagonal entries are zero. (c) Prove that a matrix is nilpotent if and only if its Jordan canonical form is nilpotent. (d)
Let J be a Jordan matrix. (a) Prove that Jk is a complete matrix for some k ≥ 1 if and only if either J is diagonal, or J is nilpotent with Jk = O. (b) Suppose that A is an incomplete matrix such that Ak is complete for some k ≥ 2. Prove that Ak = O. (A simpler version of this problem appears
Show that a real unitary matrix is an orthogonal matrix.
The Cayley-Hamilton Theorem: Let pA (λ) = det(A - λ I) be the characteristic polynomial of A. (a) Prove that if D is a diagonal matrix, then pD(D) = O. (b) Prove that if A is complete, then pA(A) = O. (c) Prove that if J is a Jordan block, then pJ(J) = O. (d) Prove that this also holds if J is a
Prove that the n vectors constructed in the proof of Theorem 8.50 are linearly independent and hence form a Jordan basis.
Prove Proposition 8.43. Proposition 8.43 If U1 and U2 are n × n unitary matrices, so is their product U1U2.
Write out a new proof of the Spectral Theorem 8.26 based on the Schur Decomposition.
A complex matrix A is called normal if it commutes with its Hermitian transpose: A†A = AAT.(a) Show that every real symmetric matrix is normal.(b) Show that every unitary matrix is normal.(c) Show that every real orthogonal matrix is normal.(d) Show that an upper triangular matrix is normal if
For each of the following Jordan matrices, identify the Jordan blocks. Write down the eigenvalues, the eigenvectors, and the Jordan basis. Clearly identify the Jordan chains.
Write down all possible 4 × 4 Jordan matrices that only have an eigenvalue 2.
Write down all possible 3 × 3 Jordan matrices that have eigenvalues 2 and 5 (and no others).
Find Jordan bases and the Jordan canonical form for the following matrices:
Choose one or more of the following differential equations, and then:(a) Solve the equation directly.(b) Write down its phase plane equivalent, and the general solution to the phase plane system.(c) Plot at least four representative trajectories to illustrate the phase portrait.(d) Choose two
(a) Write out the second order equation in terms of the coefficients a, b, c, d of the first order system.(b) Show that there is a one-to-one correspondence between solutions of the system and solutions of the scalar differential equation.(c) Use this method to solve the following linear systems,
Find the general real solution to the following systems of differential equations:(a)(b) (c) (d) (e)
Solve the following initial value problems:(a)(b) (c) (d) (e) (f) (g)
(a) Find the solution to the systemthat has initial conditions x(0) = 1, y(0) = 0. (b) Sketch a phase portrait of the system that shows several typical solution trajectories, including the solution you found in part (a). Clearly indicate the direction of increasing t on your curves.
A planar steady state fluid flow has velocity vector v = (2x - 3y, x - y)T. The motion of the fluid is described by the differential equation dx/dt = v. A floating object starts out at the point (1. 1)T. Find its position after 1 time unit.
A steady state fluid flow has velocity vector v = (-2y, 2x, z)T. Describe the motion of the fluid particles as governed by the differential equation dx/dt = v
Solve the initial value problemExplain how orthogonality can help.
(a) Find the eigenvalues and eigenvectors of(b) Verify that the eigenvectors are mutually orthogonal. Based on part (a), is K positive definite, positive semi-definite or indefinite? (c) Solve the initial value problem using orthogonality to simplify the computations.
Demonstrate that one can also solve the initial value problem in Example 9.8 by writing the solution as a complex linear combination of the complex eigen-solutions, and then using the initial conditions to specify the coefficients.
(a) Convert the third order equation(b) Solve the equation directly, and then use this to write down the general solution to your first order system. (c) What is the dimension of the solution space?
Let A be a constant matrix. Suppose u(r) solves the initial value problem Prove that the solution to the initial value problem u(t0) = b. is equal to How are the solution trajectories related?
Suppose u(f) and both solve the linear system(a) Suppose they have the same value at any one time t1. Show that they are, in fact, the same solution: for all t.(b) What happens if for some t1 ‰ t2. See Exercise 9.1.21.
Prove that the general solution to a linear system with diagonal coefficient matrix Î› = diag(Î»1,... Î»n) is given by u(t) = (c1eÎ»1t,... , cneÎ»nt)T.
(ii) Write the general solution to the systems in Exercise 9.1.13 in this form.
Find the general solution to the linear systemdu/dt = Aufor the following incomplete coefficient matrices:(a)(b) (c) (d) (e) (f) (g) (h)
Find a first order system of ordinary differential equations that has the indicated vector-valued function as a solution:(a)(b) (c) (d) (e) (f) (g) (h)
Which sets of functions in Exercise 9.1.20 can be solutions to a common first order, homogeneous, constant coefficient linear system of ordinary differential equations? If so, find a system they satisfy; if not, explain why not.In Exercise 9.1.20Determine whether the following vector-valued
Solve the third order equationby converting it into a first order system. Compare your answer with what you found in Exercise 9.1.2.

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