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

Find the Cholesky factorizations of the following matrices:a.b. c. d. e.
Find an L D VT factorization of the following symmetric matrices. Which are positive definite?a.b. c. d. e. f. g. h.
Which of the matrices in Exercise 3.5.1 have a Cholesky factorization? When valid, write out the factorization.
Write the following positive definite quadratic forms as a sum of pure squares, as in (3.71): (a) 16x21 + 25x22 (b) 5x21 - 2x1x2 + 4x22 (c) 5x21 + 4x1x2 + 3x22 (d) 3x21 -2x1x2 - 2x1x3 + 2x22 + 6x23 (e) x21 + x1x2 + x2 x3 + x23 (0) 4x21 - 2x1x2 - 4x1x3 + 1/2 x22 - x2x3 + 6x23 (g) 3x21 + 2x1x2 +
(a) For which values of c is the matrixpositive definite?(b) For the particular value c = 3, carry out elimination to find the factorization A = LDLT.(c) Use your result from part (b) to rewrite the quadratic form q(x, y, z) = x2 + 2xy + 3y2 + 2yz + Z2 as a sum of squares.(d) Explain how your
Write the following quadratic forms on M2 as a sum of squares. Which are positive definite? (a) x2 8xy + y2 (b) x2 - 4xy + 7y2 (c) x2 - 2x y - y2 (d) x2 + 6x y
Prove that the following quadratic forms on M3 are positive definite by writing each as a sum of squares: (a) x2 + 4xz + 3v2 + 5z2 (b) x2 + 3xy + 3y2 -2xz + Sz2 (c) 2x2 + x1x2 - 2x1x3 + 2x22 - 2x2x3 + 2x23
Write the following quadratic forms in matrix notation and determine if they are positive definite: (a) x2 + 4xz + 2y2 + 8yz + 12z2 (b) 3x2 - 2xy2 - 8xy + xz + z2 (c) x2 + 2xy + 2y2 -4xz - 6yz + 6z2 (d) 3x2 - x22 + 5x23 + 4x|x2 - 7x1x3 + 9x2x3 (e) x21 + 4x1x2 - 2x1x3 + 5x22 - 2x2x4 + 6x23 - x3x4
True or false;. Every planar quadratic form q(x, y) = ax2 + 2bxy + cy2 can be written as a sum of squares.
Write down a single equation that relates the five most important numbers in mathematics, which are 0, 1, e, π and i.
Prove the identities in (3.82) and (3.83).
Prove ph (z/w) = ph z - ph w = ph (z-w) is equal to the angle between the vectors representing z and w.
The phase of a complex number z = x + i y is often written as ph z = tan -1(y/x) Explain why this formula is ambiguous, and does not uniquely define phz.
Show that if we identify the complex numbers z, u; with vectors in the plane, then their Euclidean dot product is equal to Re(z w).
(a) Prove that the complex numbers z and w correspond to orthogonal vectors in R2 if and only if Re z w = 0.(b) Prove that z and i z are always orthogonal.
Prove that ez+w = ezew. Conclude that emz = (ez)m' whenever m is an integer.
(a) Use the formula e2iθ = (eiθ)2 to deduce the well-known trigonometric identities for cos 2θ and sin 2θ.(b) Derive the corresponding identities for cos 3θ and sin 3θ.(c) Write down the explicit identities for cos mθ and sinθ as polynomials in cosθ and sinθ. Hint: Apply the Binomial
Use complex exponentials to prove the identity
Prove that if z = x + I iy, then |ez| = ex, phez = y.
The formulasserve to define the basic complex trigonometric functions. Write out the formulas for their real and imaginary parts in terms of z = x + i y. and show that cos z and sin z reduce to their usual real forms when z = a- is real. What do they become when z = i y is purely imaginary?
For any integer k. prove that eki = (-1)k.
The complex hyperbolic functions are defined as(a) Write out the formulas for their real and imaginary parts in terms of z = x + iy. (b) Prove that cos i z = cos I z and sin i z = i sinhz.
Generalizing Example 2.17(c), by a trigonometric polynomial of degreein the powers of the sine and cosine functions up to degree n. degree n.(a) Use formula (3.86) to prove that any trigonometric polynomial of degree (b) Prove that any trigonometric polynomial of degree ‰¤ n can be written as a
Write the power series expansions for ei,x. Prove that the real terms give the power series for cosx, while the imaginary terms give that of sin x. Use this identification to justify Euler's formula (3.84).
The derivative of a complex-valued function f(x) = u(x) + iv(x), depending on a real variable x, is given by f'(x) = u(x) + iv(x).(a) Prove that if X = p. + i v is any complex scalar, then(b) Prove, conversely.provided Î» ‰ 0
Use the complex trigonometric formulae (3.86) and Exercise 3.6.24 to evaluate the following trigonometric integrals:a.b.c.d.How did you calculate them in first year calculus? If you're not convinced this method is easier, try the more complicated integrals (e)(f)(g)(h)
Use a computer to graph the real and imaginary parts of the complex functions z2 and 1 /z. Discuss what you see.
Use a computer to graph ph z and |z| . Discuss what you observe.
Determine whether the indicated sets of complex vectors are linearly independent or dependent.
(a) Determine whether the vectorsare linearly independent or linearly dependent.(b) Do they form a basis of C3?(c) Compute the Hermitian norm of each vector.(d) Compute the Hermitian dot products between all different pairs. Which vectors are orthogonal?(e) Do the vectors form an orthogonal or
Is the formula F = 1 valid for all complex values of z?
Find the dimension of and a basis for the following subspaces of C3:(a) The set of all complex multiples of (1. i. 1 - i)T.(b) The plane zi + iz2 + (1 - i )Z3 = 0-(c) The range of the matrix(d) The kernel of the same matrix. (e) The set of vectors that are orthogonal to (1 - i,2i, 1 + i)T.
True or false: The set of complex vectors of the fromfor c ˆˆ C forms a subspace of C2.
Find bases for the four fundamental subspaces associated with the complex matricesa.b. c.
Prove that v = x + i y and v- = x - i y are linearly independent complex vectors if and only if their real and imaginary parts x and y are linearly independent real vectors.
Prove that the space of complex m x n matrices forms a complex vector space. What is its dimension?
Let V denote the complex vector space spanned by the functions 1, eix and e~ix, where x is a real variable. Which of the following functions belong to V? (a) sinx (b) cosx-2i sinx (c) cos h x (d) sin2 1/2x (e) cos2x
Prove that the following define Hermitian inner products on C2:(a) (v, w) = v1w1 + - 2v2w-2,(b) (v, w) = v1w-1 + iv1w-2 - i v2 +2V2w2-
What is wrong with the calculation e2a π 1 = (e 2 π i)α = lα = 1?
Let A = AT be a real symmetric n x n matrix. Show that (A v) • w = v ∙ (A w) for all v, w ∈ Cn.
Let z = x + I y ∈ Cn. (a) Prove that, for the Hermitian dot product, ||z||2 = ||x||2 + ||y||2 (b) Does this formula remain valid under a more general Hermitian inner product on Cn?
Let V be a complex inner product space. Prove that, for all z, w 6 V,(a) ||z + w||2 = ||z||2 + 2 Re(z. w) + ||w||2;(b) (z, w) = 1/4(||z + xv||2 - ||z - w||2 + i ||z + i w||2 - i ||z - i w||2).
(a) How would you define the angle between two elements of a complex inner product space?(b) What is the angle between(-1,2- i, -1 +2i)Tand(-2- i.-i, 1 - i)Trelative to the Hermitian dot product?
Let 0 ≠ v ∈ Cn. Which scalar multiples cv have the same Hermitian norm as v?
Prove the Cauchy-Schwarz inequality (3.95) and the triangle inequality (3.96) for a general complex inner product. Hint: Use Exercises 3.6.8, 3.6.42.
(a) Formulate a general definition of a norm on a complex vector space.(b) How would you define analogs of the 1, 2 and oo norms on Cn?
The Hermitian adjoint of a complex m x n matrix A is the complex conjugate of its transpose, written Af = AT = AT. For example, ifthen Prove (a) (A+)+ = A, (b) (zA + uB)+ = z-A+ + w-B+ for z, w ˆˆ C, (c) (AB)+ = B+A+.
A complex matrix H is called Hermitian if it equals its Hermitian adjoint, W = H, as defined in the preceding exercise.(a) Prove that the diagonal entries of a Hermitian matrix are real.(b) Prove that (Hz) ∙ w = z ∙ (H w) forz, w ∈ C".(c) Prove that every Hermitian inner product on C" has the
(a) Write i in phase-modulus form. (b) Use this expression to find √ i.e., a complex number z such that z2 = i. Can you find a second square root? (c) Find explicit formulas for the three third roots and four fourth roots of i.
Let vt,... , v" be elements of a complex inner product space. Let K denote the corresponding n x n Gram matrix, defined in the usual manner. (a) Prove that A- is a Hermitian matrix, as defined in Exercise 3.6.49. (b) Prove that K is positive semi-definite, meaning zT K z ≥ 0 for all z ∈ C". (c)
For each the following pairs of complex-valued functions,(i) compute their L2 norm and Hermitian inner product on the interval [0, 1 ]. and then(ii) check the validity of the Cauchy-Schwarz and triangle inequalities.(a) 1, e i π x (b) x + i, x - i(c) i x2, (1 - 2i )x + 3 i
Formulate conditions on a weight function «;(*) that guarantee that the weighted integraldefines an inner product on the space of continuous complex-valued functions on [a. b].
(a) If z moves counterclockwise around a circle of radius r in the complex plane, around which circle and in which direction does w = 1/z move?(b) What about w = I?(c) What if the circle is not centered at the origin?
Show that - |z| < Rez ≤ |z| and - |z| ≤ Imz ≤ |z|.
Prove that if
(a) Explain in detail why the minimizer of ||v - b|| coincides with the minimizer of ||v - b||2.(b) Find all scalar functions F(x) for which the minimizer of F(||v - b||) is the same as the minimizer of ||v - b||.
(a) Explain why the problem of maximizing the distance from a point to a subspace does not have a solution.(b) Can you formulate a situation where maximizing distance to a point leads to a problem with a solution?
Find the closest point or points to b = ( -1.2)T that lie on (a) The a-axis (b) The y-axis (c) The line y = x (d) The line x + y = 0 (e) The line 2a + y = 0
Solve Exercise 4.1.3 when distance is measured in((a) The ∞ - norm (b) The 1 -norm
Given b ∈ R2, is the closest point on a line L unique when distance is measured in(a) The Euclidean norm?(b) The 1-norm? (c) The ∞-norm?
Let L Š‚ R2 be a line through the origin, and let b 6 R2 be any point.(a) Find a geometrical construction of the closest point x e L to b when distance is measured in the standard Euclidean norm.(b) Use your construction to prove that there is one and only one closest point.(c) Show that if 0
Suppose a and b are unit vectors in R2. Show that the distance from a to the line through b is the same as the distance from b to the line through a. Use a picture to explain why this holds. How is the distance related to the angle between the two vectors?
(a) Prove that the distance from the point (x0, y0)T to the line ax + by - 0(b) What is the minimum distance to the line Î±x + by + c = 0?
Find the minimum value of the function f(x. y, z) = x2 + 2xy + 3y2 + 2yz + z2 - 2x + 3 z + 2. How do you know that your answer is really the global minimum?
Let q(x) = xrTx be a quadratic form. Prove that the minimum value of q(x) is either 0 or - ∞.
Under what conditions does a quadratic function p(x) = xTKx-2 xTf + c have a finite global maximum? Explain how to find the maximizer and maximum value.
Why can't you minimize a complex-valued quadratic function?
For each of the following quadratic functions, determine if there is a minimum. If so, find the minimizer and the minimum value for the function.(a) x2 - 2x y + 4y2 + x - 1(b) 3x2 + 3xy + 3y2 - 2x - 2y + 4(c) x2 + 5x y + 3y2 + 2x - y(d) x2 + y2 + yz + z2 + x + y - z(e) x2 + xy - y2-yz + z2- 3(f) x2
(a) For which numbers b (allowing both positive and negative numbers) is the matrixpositive definite? (b) Find the factorization A = LDLT when b is in the range for positive definiteness. (c) Find the minimum value (depending on b it might be finite or it might be - ˆž) of the function
For each matrix K. vector f. and scalar c, write out the quadratic function p() given by (4.10). Then either find the minimizer x* and minimum value p(x'). or explain why there is none.(a)(b)(c)(d)(e)
Find the minimum value of the quadratic functionp(x .........x")for n = 2 ,3 and 4
Find the maximum value of the quadratic functions (a) -x2 + 3xv - 5y2 - x 4- 1. (b) - 2x2 4- 6xy - 3y2 + 4x - 3y.
Suppose K1 and K2 are positive definite n x n matrices. Suppose that, for i = 1. 2. The minimizer of Pi(x) = xTKix-2xTt, + ci x*. Is the minimizer of p1(x) - p2(x) + p;tx) given by x* = xj + x*2? Prove or give a counterexample.
Let K > 0. Prove that a quadratic function p(x) - xTKx - 2xTf without constant term has non-positive minimum value: p(x') ≤ 0. When is the minimum value equal to zero?
(a) Given a configuration of n points ai........a" in the plane, explain how to find the point x ‚¬ R2 that minimizes the total squared dis-(b) Apply your method when (i) a1 = (1.3), a2 = (-2,5); (ii) a1 = (0.0). a2 = (0. 1), a3 = (1.0); (iii) a1 = (0,0). a2 = (0.2). a3 = (1,2), a4 (-2,
Answer Exercise 4.3.10 when distance is measured in(a) The weighted norm ||x|| = ˆš2x21 + 3x22;(b) The norm based on the positive definite matrix
Justify the formulae in (4.26).
Find the least squares solutions to the following linear systems: (a) x + 2y = 1, 3at - y = 0, -x +2y = 3, (b) 4x-2y = 1, 2x+3y = -4, x - 2y = - 1, 2x + 2y = 2, (c) 2u + v - 2w = 1,3u-2w = 0, k-v+3w = 2, (d) x - z = - 1, 2.x - y + 3z = 1, y - 3z = 0, -5x + 2y + z = 3, (e) x1 + x2 = 2, x2 + x4 = 1.
Find the least squares solution to the linear system Ax = b whena.b. c. d. e.
LetProve, using Gaussian Elimination, that the linear system Ax = b has a unique solution. Show that the least squares solution (4.29) is the same. Explain why this is necessarily the case.
Suppose we are interested in solving a linear system Ax = b by the method of least squares when the coefficient matrix A has linearly dependent columns. Let Kx = f, where K = ATCA, f = ATCb, be the corresponding normal equations.(a) Prove that f ∈ mg K, and so the normal equations have a
Redo Exercise 4.3.1 using(a) the weighted inner product(v, w) = 2v1w1) + 4v2w2 + 3v3w3,(b) the inner product (v, w) = vTC w based on the positive definite matrix
Let b = (3,1, 2, l)7. Find the closest point and the distance from b to the following subspaces: (a) The line in the direction (1. 1. 1. I)T, (b) The plane spanned by (1,1,0,0)T and (0.0, 1, l)T, (c) The hyperplane spanned by (1,0,0,0)T, (0. 1.0. 0)T, (0, 0, l,0)T, (d) The hyperplane defined by the
Find the closest point and the distance from b = (1,1,2, -2)T to the subspace spanned by (1.2. - 1.0)T, (0. 1,-2, -l)T, (1,0,3, 2)T.
Redo Exercises 4.3.4,4.3.5 using(i) The weighted inner product(v. w) = 1/5 v1w1) + v2w2 + 1/2 v3w3 + V4w4(ii) The inner product based on the positive definite matrix
Let A be an m x n matrix with rank A = n.(a) Prove that the matrix P = A(ATA)~l AT is a projection matrix, meaning that P2 = P, cf. Exercise 2.5.8.(b) Construct the projection matrix corresponding to(i)(ii)(iii)(iv)(c) Prove that P is symmetric.(d) Prove that mg P = mg A.(e) Show that v* = P b is
Find the straight line y = a + f) t that best fits the following data in the least squares sense:(a).(b). (c)
Show, by constructing explicit examples, that -t2 (t)-2 and Ty ≠ t-y-. Can you find any data for which either equality is valid?
Given points t1,........tm provehereby justifying (4.38).
For the following data values, construct the interpolating polynomial in Lagrange form:(a)(b)(c)(d)(e)
Find and graph the polynomial of minimal degree that passes through the following points: (a) (3,-1), (6, 5); (b) (0,6), (-2,4), (1,10); (c) (-2, 3), (0,-1), (1,-3); (d) (0,-1), (1,0), (-1,2), (2,-1); (e) (0, -3), (-1.-3), (1, -1), (-2, 17), (2, 9).
Given(a) Find the best straight line y = a + fit that fits the data in the least squares sense; (b) Find the best parabola y - Î± + Î²t + yt2 that fits the data. Interpret the error.
A student runs an experiment six times in an attempt to obtain an equation relating two physical quantities x and y. For x = 1,2,4, 6, 8, 10 units, the experiments result in corresponding y values of 3, 3, 4, 6, 7, 8 units. Find and graph the following:(a) The least squares line;(b) The least
(a) Write down the Taylor polynomials of degrees 2 and 4 at t = 0 for the function f(t) = et.(b) Compare their accuracy with the interpolating and least squares polynomials in Examples 4.14 and 4.18.
Given the known values of sin t at t = 0°, 30°, 45°, 60°, find the following approximations:(a) The least squares linear polynomial,(b) The least squares quadratic polynomial,(c) The quadratic Taylor polynomial at t = 0,(d) The interpolating polynomial.(e) The cubic Taylor polynomial at t =
The proprietor of an internet travel company compiled the following data relating the annual profit of the firm to its annual advertising expenditure (both measured in thousands of dollars):(a) Determine the equation of the least squares line. (b) Plot the data and the least squares line. Estimate
(a) Find the least squares linear polynomial approximating √t on [0, 1]. choosing six different exact data values.(b) How much more accurate is the least squares quadratic polynomial based on the same data?
Find the quartic (degree 4) polynomial that exactly interpolates the function tan t at the five data points t0 = 0, t1 = .25. t2 = -5, t3 = .75, t4 = 1. Compare the graphs of the two functions over 0 ≤ t ≤ 1/2π.
A table of logarithms contains the following entries:Approximate log,0e by constructing an interpolating polynomial of (a) Degree two using the entries at x = 1.0, 2.0, and 3.0, (b) Degree three using all the entries
Let q(t) denote the quadratic interpolating polynomial that goes through the data points (to, Vo), (t1. .V1), (t2, y2).(a) Under what conditions does q(t) have a minimum? A maximum?(b) Show that the minimizing/maximizing value is atWhere What is q(t')?

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