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linear algebra

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

Let U and V be subspaces or Rn. In the case that U ∩ V = {0} we have the following dimension relation dim (U + V) = dim U + dim V (See Exercise 18 in Section 4 of Chapter 3.) Make use of the result from Exercise 13 to prove the more general theorem dim (U + V)= dim U + dim V - dim({/ n V)
Repeat Exercise 8 using the modified Gram-Schmidt process and compare answers.In Exercise 8Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R4 spanned by x1 = (4, 2, 2)T, x2 = (2,0,0,2)T, and x3 = (1, 1,-1, l)T.
Prove each of the following.(a) H'n(x) = 2nHn-1(x), n = 0. 1...(b) Hʹʹn(x) -2xHʹn(x) + 2nHn(x) = 0. n = 0. I,...
Show that if f(x) is a polynomial of degree less than n then f(x) must equal the interpolating polynomial P(x) in (7), and hence the sum in (7) gives the exact value for ∫ba f(x)w(x)dx.
Let x1, x2, ... , xn be distinct points in the interval [-1, 1] and letwhere the Li's are the Lagrange functions for the points x1, x2, ... , xn. (a) Explain why the quadrature formula will yield the exact value of the integral whenever f(x) is a polynomial of degree less than n. (b) Apply the
Let x1, x2, ..., xn be the roots of the Legendre polynomial Pn. If the Ai's are defined as in Exercise 15, then the quadrature formulawill be exact for all polynomials of degree less than 2n. (a) If 1 Pj(x1)A1 + Pj(x2)A2 + ........ + Pj(xn)An = (l, Pj) = 0 (b) Use the results from parts (a) and (b)
Let Q0(x), Q1(x), ... be an orthonormal sequence of polynomials, that is, it is an orthogonal sequence of polynomials and ||Qk|| = 1 for each k.(a) How can the recursion relation in Theorem 5.7.2 be simplified in the case of an orthonormal sequence of polynomials?(b) Let A be a root of Qn. Show
Show that the Chebyshev polynomials have the following properties: (a) 2Tm(x)Tn(x) = Tm+n(x) + Tm-n(x), for m > n (b) Tm(Tn(x)) = Tmn(x)
Let p0, p1,... be a sequence of orthogonal polynomials and let an denote the lead coefficient of pn. Prove that ||pn||2 = an(xn, pn)
Let Tn(x) denote the Chebyshev polynomial of degree n and defineun-1(x) = 1/n TÊ¹n(x)If x = cos Î¸, show that
Let Un-1(x) be defined as in Exercise 6 for n > 1 and define U-1(x) = 0. Show that (a) Tn(x) = Un(x) - xUn-1(x), for n > 0 (b) Un{x) = 2xUn-1(x) - Un-2(x), for n > 1
Show that the Ui's defined in Exercise 6 are orthogonal with respect to the inner productThe Uj's are called Chebyshev polynomials of the second kind.
For n =0, 1,2, show that the Legendre polynomial Pn(x) satisfies the second-order equation (1 - x2)y" - 2xy' + n(n + l)y = 0
Show that the matrixwill have complex eigen values if is not a multiple Ï€. Give a geometric interpretation of this result.
Let A = (aij be an n Ã— n matrix with elgenvalues Î»1, ... ,Î»n. Show that
Let A be a 2 x 2 matrix and let p(λ) = λ2 + bλ + c be ihc characteristic polynomial of λ. Show that b = -tr(A) and c = dct(A).
Let λ be a nonzero elgenvalue of A and let x be an eigenvector belonging to λ. Show that Amx is also an eigenvector belonging to λ form = 1,2,....
Let A be an n × n matrix. Show that a vector x in Rn is an elgenvector belonging to A if and only if the subspace S of Rn spanned by x and Ax has dimension 1.
Let a = a + bi and Î² = c + di be complex scalars and let A and B be matrices withcomplex entries.(a) Show that (b) Show that the (i. j) entries of AB- and A-B- arc equal and hence thatA-B- = A-B-
Let Q be an orthogonal matrix.(a) Show that if A is an eigen value of Q then |λ| = I.(h) Show that | det (Q)| = 1.
Let Q be an orthogonal matrix with an eigen value λ1 = I and lel x be an eigenvector belonging to A1. Show that x is also an eigenvector of QT
Show that the eigen values of a triangular matrix are the diagonal elements of the matrix.
Let Q be 3 x 3 orthogonal matrix whose determinant is equal to 1.(a) If the eigenvalues of Q are all real and if they an' ordered so that λ1 ≥ λ2 ≥ λ3. deternine values of all possible triples of eigenvaIues (λ1. λ2, λ3).(b) In the case that the eigenvalues λ2 and λ3. are complex. what
Let x1 .... xr be eigenvectors of an n x n matrix A and let S be the ssubspace of R spanned by x1. x2 ....xf Show that S is invariant under A (i.e.. show that Ax ∈ S whenever x ∈ 5).
Let B = S-1 AS and let x be an eigenvector of B belonging to an eigenvalue λ show that S is an eigenvector of A belonging to λ.
Show that if two n × n matrices A and B have a common eigenvector x (but not necessarily a common eigenvalue), then x will also be an eigenvector of any matrix of the form C αA + βb.
Let A be an n x n matrix and let A he a nonzero elgenvalue of A. Show that if x is an eigene.1or belonging to A, then x is in the column space of A. Hence the eigenspace corresxponding to λ is a subspace of the column space of A.
Let (u1. u2, . . . , un), be an orthonornial basis for Rn and let A be a linear combinationof the rank I matrices u1uT1.u2uT2..... unun. Ifhow that A is a symmetric matrix with eigenvalues c1,c2 and that cn, is an eigenvector belonging to ci, for each i.
Let A be a matrix whose columns all add up to a fixed constant δ. Show that δ is an elgenvalue of A.
Let λ1 and λ2 be distinct eigenvaIues of A. Let x be an cigenvector of A belonging to λ1 and let y he an elgenvector of AT belonging to λ2. Show that x and y are orthogonal.
Let A and B be n × n is matrices. Show that (a) If λ is a noniem eigenvalue of AB. then i is also an eigenvalue of BA. (b) If A = 0 is an eigenvalue of Ab, then λ = 0 is also an eigenvalue of BA.
Prove that there do not exist n × n matrices A and B such that AB - BA = 1
Let A be an n × n matrix. Prove that A is singular if and only if λ = 0 is an eigen value of A.
Let p(Î») = (-1 )n(Î»n - Î±n-1Î»n-1 -....- Î±1Î» - Î±o) he a polynomial of degree n ‰¥ I(a) Show that if Î»i, is a root of p(Î») = 0 then Î»i is an elgenvalue of C with eigenvector x = (Î»in-1,,Î» in-2...... Î»i,1)T ,(b) Use part (a) to show that if p(Î») has n
The result given in Exercise 30b) holds even if all (he eigenvalues of p(Î») are not distinct. Prove this as follows:(a) Letand use mathematical induction to prove that det( Dm(Î»)) = (-1)m + Î±mÎ»m + Î±m-1Î»m-1 + ....... +
Let A be a nonsingular matrix and let λ be an elgenvalue of A. Show that 1/λ is an eigenvalue of A-1.
Let λ be an cigenvalue of A and let x be an eigenvecior belonging to λ. Use mathematical induction to show that λm is an cigenvalue of AM and x is an eigenvector of Am belonging to Am for in = 1, 2
An n x n matrix A is said to be a idempotent if A2 = A. Show that if λ is an eigenvalue of an idempotent matrix, then λ must be either 0 or I.
A n × n matrix is said to be nilpotent if Ak = 0 for some positive integer k. Show that all eigen values of a nilpotent matrix are 0.
Show that A and AT have the same eigen values. Do they necessarily have the same eigenvectors? Explain.
Transform the ,ith-order equationinto a system of first-order equations by setting v = v and yj = for y;j-1 = 2, ... , n. Determine the characteristic polynomial of the coefficient matrix of this system.
Givenis the solution to the initial value problem Y = AY. Y(0) = Y0 (a) Show that (b) Let x = (x1,...,x,,)and c = (c1,...,cn)T. Assurning that the vector x1...,xn are linearly independent, show that c X-1Y0.
In Application 2. assume that the solutions are of the form x1 = α1 sin σt x2 = a sin fl. Substitute these expressions into he system and solve for (he frequency and the amplitudes α1 and α2.
Two masses are connected by springs as shown in the diagram. Both springs have the same spring constant, and the end of the first spring is fixed. If x1 and x2 represent the displacements from the equilibrium position. derive a system of second-order differential equations that describes the motion
In each of the following, factor the matrix A into a product XDX-1. where D is diagonal.a.b.c.d.e.f.
Let A he an n x n matrix with positive real cigenvalucs Î»1 > Î»2 >...... >Î»n. Let xi he an eigenvector belonging to Î»i, for each i, and let x = a1x1 +........ + anxn.a.b.
Let A be an n x n matrix with an eigenvalue A of multiplicity n. Show that A diagonalizable if and only if A = λl.
Let A be a diagonalizable matrix and let X be the diagonalizing matrix. Show that the column vectors of X that corresnd to nonzero cigenvalues of A form a basis for R(A.
Let A be an n x n matrix and let A be an eigenvalue of A whose eigenspace has dimension k. where Iwhere I is the k x k identity mairk.(b) Use Theorem 6.1.1 to show that Î» is an eigenaIue of A with multiplicity it least k.
Let x. y be nonzero sectors in Rn. n ≥ 2. and let A = xyT Show that(a) A = 0 is an eigenvalue of A with n - I linearly independent elgenvectors and consequently has multiplicity at least n - 1 (see Exercise 15).(b) The remaining elgenvalue of A isλn = tr A= xTyand x is an eigenvector belonging
Let A be a diagonalizable n x n matrix. Prove that if B is any matrix that is similar to A. then B is diagonalizable.
Show that if A and B are two n x n matrices that both have the same diagonalizing matrix X. then AB = BA.
Let T be an upper triangular matrix with distinct diagonal entries (i.e., tii ≠ tjj, whenever I ≠ j). Show that there is an upper triangular matrix R that diagonalizes T.
For each of the matrices in Exercise I, use the XDX-1 - factorization to compute A6.
Let A be an ii x n stochastic matrix and let e be the vector in Rn whose entries are all equal to I. Show that c is an eigenvector of AT. Explain why a stochastic matrix must have A = 1 as an eigenvalue.
The transition matrix in Example 5 has the property that both its rows and its columns all add up to I. In general. a matrix A is said to be doubly stochastic if both A and AT are stochastic. Let A be an ii x ii doubly stochastic matrix whose eigenvalues satisfy λ1 = 1 and |λ1| < 1 for j =
Let A be the PageRank transition matrix and let xk be a vector in the Markov chain with starting probability vector x0. Since n is very large, the direct multiplication xk+1 is computationally intensive. However, the computation can be simplified dramatically if we take advantage of the structured
Use the definition of the matrix exponential to compute e for each of the following matrices:(a) (b)
In each of the following, solve the initial value problem Y' = AY, Y(O) = Y0 by computing etA y0.
Let λ be an eigenvalue of an n x n matrix A. and let x be an eigenveclor belonging to λ. Show that eλ is an eigenvalue of eA and x is an eigenvector of eA belonging to eλ.
For each of the nonsingular matrices in Exercise I, use the XDX-1 factorization to compute A-1
Show that eA is nonsingular for any diagonalizable matrix A.
Let A be a diagonalizable matrix with characteristic polynomial(a) If D is a diagonal matrix whose diagonal entries are the eigenvalues of A, show that = p(D) + a1Dn + a2Dn-1 + an+1I = 0 (b) Show that p(A) = O. (c) Show that if a+ 0 ‰ , then A is nonsingular and A-1 = q(A) for some
For each of the following, find a matrix B such that B2 = A.a.b.
Let A be a nondefective n x n matrix with diagonalizing matrix X. Show that the matrix Y = (X-1)T diagonalizes AT.
Let A be a diagonalizable matrix whose eigenvalues are all either I or -1. Show that = A.
Show that any 3 Ã—3 matrix of the formIs defective.
For each of the following. find all possible values of the scalar a that make the matrix defective or show that no such values exist.(a)
Let A be a 4 x 4 marix and let λ he an eigenvalue of multiplicity 3. If A - λl has rank 1. is A defective? Explain.
Let {u1,..., un} be an orthonormal basis for a complex inner product space V, and letz = a1u1 + a2u2 + ... + anunw = b1u1 + b2u2 + ... + bnunShow that
Givenfind a matrix B such that BHB = A.
Let U be a unitary matrix. Prove: (a) U is normal. (b) If λ is an eigenvalue of U, then |λ| = 1.
Show that if a matrix U is both unitary and Hermitian then any eigenvalue of U must equal either 1 or -1.
Let A be a 2 × 2 matrix with Schur decomposition UTUH and suppose that t12 ≠ 0. Show that (a) The eigenvalues of A are λ1 = t11 and λ2 = t22. (b) u1 is an eigenvector of A belonging to λ1 = t11. (c) u2 is not an eigenvector of A belonging to λ2 = t22.
Let A be a n × n matrix with Schur decomposition UTUH. Show that if the diagonal entries of T are all distinct, then there is an upper triangular matrix R such that X = UR diagonalizes A.
Show that M = A + iB (A and B real matrices) is skew Hermitian if and only if A is skew symmetric and B is symmetric.
Show that if A is skew Hermitian and λ is an eigenvalue of A then λ is purely imaginary (i.e., λ = bi, where b is real).
Show that if A is a normal matrix then each of the following matrices must also be normal.(a) AH(b) I + A(c) A2
Let(a) Show that {z1, z2} is an orthonormal set in C2.
Let A be a real 2 Ã— 2 matrix with the property that a21a12 > 0, and letCompute B = SAS-1. What can you conclude about the eigenvalues and eigenvectors of B? What can you conclude about the eigenvalues and eigenvectors of A? Explain.
Let p(x) = -x3 + cx2 + (c + 3)x + 1, where c is a real number. Let C denote the companion matrix of p(x),and let (a) Compute A-1CA. (b) Use the result from part (a) to prove that p(x) will have only real roots regardless of the value of c.
Let A be a Hermitian matrix with eigenvalues λ1,..., λn and orthonormal eigenvectors u1,..., un. Show that A = λ1u1u1H + λ2u2u2H + ... + λnununH
Let A be a Hermitian matrix with eigenvalues Î»1 ‰¥ Î»2 ‰¥ ... ‰¥ Î»n and orthonormal eigenvectors u1,...,un. For any nonzero vector x in Cn the Rayleigh quotient p(x) is defined by(a) If x = c1u1 + ... + cnun show that (b) Show
Find an orthogonal or unitary diagonalizing matrix for each of the following:(a)(b) (c) (d) (e) (f) (g)
Show that the diagonal entries of a Hermitian matrix must be real.
Let A and C be matrices in CmÃ—n and let B ˆˆ CnÃ—r. Prove each of the following rules:(a) (AH)H = A(b)
Show that (z, w) = wHz defines an inner product on Cn.
Let x, y, z be vectors in Cn and let α and β be complex scalars. Show that (z, αx + βy) = (z, x) + (z, y)
Show that A and AT have the same nonzero singular values. How are their singular value decompositions related?
Let A be an m Ã— n matrix of rank n with singular value decomposition Uˆ‘VT. Let ˆ‘+ denote the n Ã— m matrixand define A+ = Vˆ‘+ UT. Show that = A+b satisfies the normal equations ATAx = ATb.
Let A+ be defined as in Exercise 10 and let P = AA+. Show that P2 = P and PT = P.Exercise 10Let A be an m Ã— n matrix of rank n with singular value decomposition Uˆ‘VT. Let ˆ‘+ denote the n Ã— m matrixand define A+ = Vˆ‘+ UT. Show that = A+b
For each of the matrices in Exercise 2:(a) Determine the rank.(b) Find the closest (with respect to the Frobenius norm) matrix of rank 1.Exercise 2(a)(b) (c)
The matrixhas singular value decomposition Use the singular value decomposition to find orthonormal bases for R(A) and N(AT).
Prove that, if A is a symmetric matrix with eigenvalues λ1, λ2,..., λn, then the singular values of A are |λ1|, |λ2|,..., |λn|.
Let A be an m × n matrix with singular value decomposition U∑VT, and suppose that A has rank r, where r < n. Show that {v1, ..., vr} is an orthonormal basis for R(AT).
Let A be an n × n matrix. Show that ATA and AAT are similar.
Show that if σ is a singular value of A then there exists a nonzero vector x such that σ = ||Ax||2 / ||x||2
Let A be a singular n × n matrix. Show that ATA is positive semidefinite, but not positive definite.
Let A be a symmetric n Ã— n matrix with eigenvalues Î»1,..., Î»n. Show that there exists an orthonormal set of vectors {x1, ..., xn} such that
Let A be a symmetric positive definite n × n matrix and let S be a nonsingular n × n matrix. Show that STAS is positive definite.

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