- 5.B.1: Suppose T 2 L.V / and there exists a positive integer n such that T...
- 5.B.2: Suppose T 2 L.V / and .T 2I /.T 3I /.T 4I / D 0. Suppose isan eigen...
- 5.B.3: Suppose T 2 L.V / and T 2 D I and 1 is not an eigenvalue of T. Prov...
- 5.B.4: Suppose P 2 L.V / and P2 D P. Prove that V D nullP rangeP.
- 5.B.5: Suppose S;T 2 L.V / and S is invertible. Suppose p 2 P.F/ is apolyn...
- 5.B.6: Suppose T 2 L.V / and U is a subspace of V invariant under T. Prove...
- 5.B.7: Suppose T 2 L.V /. Prove that 9 is an eigenvalue of T 2 if and only...
- 5.B.8: Give an example of T 2 L.R2/ such that T 4 D 1.
- 5.B.9: Suppose V is finite-dimensional, T 2 L.V /, and v 2 V with v 0.Let ...
- 5.B.10: Suppose T 2 L.V / and v is an eigenvector of T with eigenvalue .Sup...
- 5.B.11: Suppose F D C, T 2 L.V /, p 2 P.C/ is a polynomial, and 2 C.Prove t...
- 5.B.12: Show that the result in the previous exercise does not hold if C is...
- 5.B.13: Suppose W is a complex vector space and T 2 L.W / has no eigenvalue...
- 5.B.14: Give an example of an operator whose matrix with respect to some ba...
- 5.B.15: Give an example of an operator whose matrix with respect to some ba...
- 5.B.16: Rewrite the proof of 5.21 using the linear map that sends p 2 Pn.C/...
- 5.B.17: Rewrite the proof of 5.21 using the linear map that sends p 2 Pn2 ....
- 5.B.18: Suppose V is a finite-dimensional complex vector space and T 2 L.V ...
- 5.B.19: Suppose V is finite-dimensional with dim V >1 and T 2 L.V /. Provet...
- 5.B.20: Suppose V is a finite-dimensional complex vector space and T 2 L.V ...
Solutions for Chapter 5.B: Eigenvectors and Upper-Triangular Matrices
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition
Tv = Av + Vo = linear transformation plus shift.
Upper triangular systems are solved in reverse order Xn to Xl.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
peA) = det(A - AI) has peA) = zero matrix.
Column space C (A) =
space of all combinations of the columns of A.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·