Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Write A as M + N, symmetric matrix plus skew-symmetric matrix: (MT = M, NT = -N). For any square matrix, M = A"';AT and N = __ add up to A

Introduction to Linear Algebra 4
If C is symmetric prove that ATCA is also symmetric. (Transpose it.) When A is 6 by 3, what are the shapes of C and AT CA?

Introduction to Linear Algebra 4
Find the eigenvalues and the unit eigenvectors of [ 2 2 2] A= 2 0 0 . 200

Introduction to Linear Algebra 4
Find an orthogonal matrix Q that diagonalizes A = [-~ ~ ]. What is A?

Introduction to Linear Algebra 4
Find an orthogonal matrix Q that diagonalizes this symmetric matrix: [ 1 0 2] A = 0 -1 -2 . 2 -2 0

Introduction to Linear Algebra 4
Find all orthogonal matrices that diagonalize A = [1 ~ 12] 16 .

Introduction to Linear Algebra 4
(a) Find a symmetric matrix [~ ~] that has a negative eigenvalue. (b) How do you know it must have a negative pivot? (c) How do you know it can't have two negative eigenvalues?

Introduction to Linear Algebra 4
If A 3 = 0 then the eigenvalues of A must be . Give an example that has A =1= O. But if A is symmetric, diagonalize it to prove that A must be zero.

Introduction to Linear Algebra 4
If)" = a + ib is an eigenvalue of a real matrix A,..Qten its conjugate A = a - ib is also an eigenvalue. (If Ax = AX then also Ax = AX.) Prove that every real 3 by 3 matrix has at least one real eigenvalue.

Introduction to Linear Algebra 4
Here is a quick "proof" that the eigenvalues of all real matrices are real: False proof Ax = AX gives x T Ax = AX T X xTAx so ).. =-- xTx is real.

Introduction to Linear Algebra 4
Write A and B in the form AIX IX T + A2x2x1 of the spectral theorem QAQT: 12] 16 (keep IIxIlI = IIx211 = 1).

Introduction to Linear Algebra 4
Every 2 by 2 symmetric matrix is AlxIxI + A2X2XI = AlP! + A2P2. Explain PI + P2 = xIxT + x2xI = I from columns times rows of Q. Why is PIP2 = O?

Introduction to Linear Algebra 4
What are the eigenvalues of A = [_g ~]? Create a 4 by 4 skew-symmetric matrix (AT = - A) and verify that all its eigenvalues are imaginary.

Introduction to Linear Algebra 4
(Recommended) This matrix M is skew-symmetric and also Then all its eigenvalues are pure imaginary and they also have I A I = 1. (II M x II = II x II for every x so IIAx II = IIx II for eigenvectors.) Find all four eigenvalues from the trace of M: 0 1 1 1 M=- 1 -1 0 -1 1 -1 1 0 -1 can only have eigen

Introduction to Linear Algebra 4
Show that A (symmetric but complex) has only one line of eigenvectors: A = [~ _~] is not even diagonalizable: eigenvalues A = 0, O. AT = A is not such a special property for complex matrices. The good property is AT = A (Section 10.2). Then all A'S are real and eigenvectors are orthogonal.

Introduction to Linear Algebra 4
Even if A is rectangular, the block matrix B = [1T ~] is symmetric: Bx = AX is which is Az =AY ATy = AZ. (a) Show that -A is also an eigenvalue, with the eigenvector (y, -z). (b) Show that AT Az = A 2 Z , so that A 2 is an eigenvalue of AT A. (c) If A = ! (2 by 2) find all four eigenvalues and eigenv

Introduction to Linear Algebra 4
If A = [}] in Problem 16, find all three eigenvalues and eigenvectors of B.

Introduction to Linear Algebra 4
1. Suppose Ax = AX and Ay = Oy and A f:. O. Then y is in the nullspace and X is in the column space. They are perpendicular because . Go carefully-why are these subspaces orthogonal? 2. If Ay = {3 y, apply this argument to A - {3!. The eigenvalue of A - {3! moves to zero and the eigenvectors stay the

Introduction to Linear Algebra 4
Find the eigenvector matrix S for A and for B. Show that S doesn't collapse at d = 1, even though A = 1 is repeated. Are the eigenvectors perpendicular? [ -d 0 B = 0 1 o 0 have A = 1, d, -d.

Introduction to Linear Algebra 4
Write a 2 by 2 complex matrix with AT = A (a "Hermitian matrix"). Find A} and A2 for your complex matrix. Adjust equations (1) and (2) to show that the eigenvalues of a Hermitian matrix are real.

Introduction to Linear Algebra 4
True (with reason) or false (with example). "Orthonormal" is not assumed. (a) A matrix with real eigenvalues and eigenvectors is symmetric. (b) A matrix with real eigenvalues and orthogonal eigenvectors is symmetric. (c) The inverse of a symmetric matrix is symmetric. (d) The eigenvector matrix S of

Introduction to Linear Algebra 4
(A paradox for instructors) If AAT = AT A then A and AT share the same eigenvectors (true). A and AT always share the same eigenvalues. Find the flaw in this conclusion: They must have the same S and A. Therefore A equals AT.

Introduction to Linear Algebra 4
(Recommended) Which of these classes of matrices do A and B belong to: Invertible, orthogonal, projection, permutation, diagonalizable, Markov? I I I l] Which of these factorizations are possible for A and B: LU, QR, SAS-1, QAQT?

Introduction to Linear Algebra 4
What number bin [i~] makes A = QAQT possible? What number makes A = SAS-1 impossible? What number makes A-I impossible?

Introduction to Linear Algebra 4
Find all 2 by 2 matrices that are orthogonal and also symmetric. Which two numbers can be eigenvalues?

Introduction to Linear Algebra 4
This A is nearly symmetric. But its eigenvectors are far from orthogonal: [ 1 10-15 ] A = 0 1+10-15 has eigenvectors and [7] What is the angle between the eigenvectors?

Introduction to Linear Algebra 4
(MATLAB) Take two symmetric matrices with different eigenvectors, say A = [A g] and B = [r AJ. Graph the eigenvalues AI(A +tB) andA2(A +tB) for-8 < t < 8. Peter Lax says on 'page 113 of Linear Algebra that Al and A2 appear to be on a collision course at certain values of t. "Yet at the last minute th

Introduction to Linear Algebra 4
For complex matrices, the symmetry AT = A that produces real eigenvalues changes to AT = A. From det(A - AI) = 0, find the eigenvalues of the 2 by 2 "Hermitian" matrix A = [4 2 + i; 2 - i 0] = AT. To see why eigenvalues are real when AT = A, adjust equation (1) ofthe text to A x = A x. Transpose to x

Introduction to Linear Algebra 4
Normal matrices have A A = AA . For real matrices, A A = AA includes symmetric, skew-symmetric, and orthogonal. Those have real A, imaginary A, and IAI = 1. Other normal matrices can have any complex eigenvalues A. Key point: Normal matrices have n orthonormal eigenvectors. Those vectors Xi probably

Introduction to Linear Algebra 4
If Amax is the largest eigenvalue of a symmetric matrix A, no diagonal entry can be larger than Amax. What is the first entry all of A = QAQT? Show why all < Amax

Introduction to Linear Algebra 4
Suppose AT = -A (real antisymmetric matrix). Explain these facts about A: (a) x TAx = 0 for every real vector x. (b) The eigenvalues of A are pure imaginary. (c) The determinant of A is positive or zero (not negative). For (a), multiply out an example of x T Ax and watch terms cancel. Or reverse xT(A

Introduction to Linear Algebra 4
If A is symmetric and all its eigenvalues are A = 2, how do you know that A must be 21? (Key point: Symmetry guarantees that A is diagonalizable. See "Proofs of the Spectral Theorem" on web.mit.edu/18.06.)