- 5.SE.1E: Mark each statement as True or False. Justify each answer.a. If A i...
- 5.SE.2E: Show that if x is an eigenvector of the matrix product AB and Bx ? ...
- 5.SE.3E: Suppose x is an eigenvector of A corresponding to an eigenvalue a. ...
- 5.SE.4E: Use mathematical induction to show that if is an eigenvalue of Am, ...
- 5.SE.5E: If p (t) = c0 + c1t + c22 + + ct, define p (A)to be the matrix form...
- 5.SE.6E: a. Let B = 5I 2. Show that B is diagonalizable by finding a suitabl...
- 5.SE.7E: Suppose A is diagonalizable and p (t) is the characteristic polynom...
- 5.SE.8E: a. Let A be a diagonalizable n × n matrix. Show that if the multipl...
- 5.SE.9E: Show that I – A is invertible when all the eigenvalues of A are les...
- 5.SE.10E: Show that if A is diagonalizable, with all eigenvalues less than 1 ...
- 5.SE.11E: Let u be an eigenvector of A corresponding to an eigenvalue and let...
- 5.SE.12E: Use formula (1) for the determinant in Section 5.2 to explain why d...
- 5.SE.13E: Use Exercise 12 to find the eigenvalues of the matrices in Exercise...
- 5.SE.14E: Use Exercise 12 to find the eigenvalues of the matrices in Exercise...
- 5.SE.15E: Let J be the n × n matrix of all 1’s, and consider Use the results ...
- 5.SE.16E: Apply the result of Exercise 15 to find the eigenvalues of the matr...
- 5.SE.17E: Let Recall from Exercise 25 in Section 5.4 that trA (the trace of A...
- 5.SE.19E: Exercises 19–23 concern the polynomialp (t) = 0 + a1t + + an + tnan...
- 5.SE.20E: Exercises 19–23 concern the polynomialp( t ) = a0 + a1t + + an-1tn-...
- 5.SE.21E: Exercises 19–23 concern the polynomialp (t) = a0 + a1t + + an-1tn-1...
- 5.SE.22E: Exercises 19–23 concern the polynomial and an n × n matrix Cp calle...
- 5.SE.23E: Exercises 19–23 concern the polynomial and an n × n matrix Cp calle...
- 5.SE.24E: [M] The MATLAB command roots(p) computes the roots of the polynomia...
- 5.SE.25E: [M] Use a matrix program to diagonalize if possible. Use the eigenv...
- 5.SE.26E: [M] Repeat Exercise 25 for Reference:[M] Use a matrix program to di...
Solutions for Chapter 5.SE: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications | 4th Edition
peA) = det(A - AI) has peA) = zero matrix.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Outer product uv T
= column times row = rank one matrix.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).