Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Find the norms II A II = Amax and condition numbers c = Amaxl Amin of these positive definite matrices

Introduction to Linear Algebra 4
Find the norms and condition numbers from the square roots of Amax(AT A) and Amin(AT A). Without positive definiteness in A, we go to AT A !

Introduction to Linear Algebra 4
Explain these two inequalities from the definitions (3) of II A II and II B II:

Introduction to Linear Algebra 4
Use IIAA-111 < IIAII IIA- 1 II to prove that the condition number is at least 1.

Introduction to Linear Algebra 4
Why is I the only symmetric positive definite matrix that has Amax = Amin = I? Then the only other matrices with II A II = I and II A-III = 1 must have AT A = I. Those are matrices: perfectly conditioned.

Introduction to Linear Algebra 4
Orthogonal matrices have norm II Q II = 1. If A = Q R show that II A II < II R II and also IIRII < IIAII. Then IIAII = IIQII IIRII. Find an example of A = L U with IIAII < IILIIIIUII

Introduction to Linear Algebra 4
(a) Which famous inequality gives II (A + B)x II < IIAx II + IIBx II for every x? (b) Why does the definition (3) of matrix norms lead to IIA + BII < IIAII + IIBII?

Introduction to Linear Algebra 4
Show that if A is any eigenvalue of A, then I A I < II A II. Start from Ax = AX.

Introduction to Linear Algebra 4
The "spectral radius" peA) = IAmaxl is the largest absolute value of the eigenvalues. Show with 2 by 2 examples that peA + B) < peA) + pCB) and p(AB) < p(A)p(B) can both be false. The spectral radius is not acceptable as a norm.

Introduction to Linear Algebra 4
(a) Explain why A and A-I have the same condition number. (b) Explain why A and AT have the same norm, based on A(AT A) and A(AAT)

Introduction to Linear Algebra 4
Estimate the condition number of the ill-conditioned matrix A = [~ 1.0\01]'

Introduction to Linear Algebra 4
Why is the determinant of A no good as a norm? Why is it no good as a condition number?

Introduction to Linear Algebra 4
(Suggested by C. Moler and C. Van Loan.) Compute b - Ay and b - Az when b = [.217] A = [.780 .563] Y = [ .341] z = [ .999] . . 254 .913 .659 -.087 -1.0 Is y closer than z to solving Ax = b? Answer in two ways: Compare the residual b-Ay tob-Az. Then compare y andztothetruex = (1,-1). Both answers can

Introduction to Linear Algebra 4
(a) Compute the determinant of A in Problem 13. Compute A-I. (b) If possible compute II A II and II A -111 and show that c > 106

Introduction to Linear Algebra 4
Problems 15-19 are about vector norms other than the usuallix II = ~.

Introduction to Linear Algebra 4
Problems 15-19 are about vector norms other than the usuallix II = ~.

Introduction to Linear Algebra 4
Problems 15-19 are about vector norms other than the usuallix II = ~.

Introduction to Linear Algebra 4
Problems 15-19 are about vector norms other than the usuallix II = ~.

Introduction to Linear Algebra 4
Problems 15-19 are about vector norms other than the usuallix II = ~.

Introduction to Linear Algebra 4
The eigenvalues of the -1, 2, -1 difference matrix K are A = 2 - 2 cos (j Jr In + 1). Estimate Amin and,Amax and c = cond(K) = Amax/Amin as n increases: c ~ Cn2 with what constant' C? Test this estimate with eig(K) and cond(K) for n = 10,100,1000

Introduction to Linear Algebra 4
For unsymmetric matrices, the spectral radius p = max IAi I is not a norm (Problem 9). But still IIAn II grows or decays like pn for large 11. Compare those numbers for A = [1 1; 0 1.1] using the command norm. In particular A n ~ 0 when p < 1. This is the key to Section 9.3 with A = S-I T.