 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 finitedimensional, 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 finitedimensional complex vector space and T 2 L.V ...
 5.B.19: Suppose V is finitedimensional with dim V >1 and T 2 L.V /. Provet...
 5.B.20: Suppose V is a finitedimensional complex vector space and T 2 L.V ...
Solutions for Chapter 5.B: Eigenvectors and UpperTriangular Matrices
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics)  3rd Edition
ISBN: 9783319110790
Solutions for Chapter 5.B: Eigenvectors and UpperTriangular Matrices
Get Full SolutionsSince 20 problems in chapter 5.B: Eigenvectors and UpperTriangular Matrices have been answered, more than 5770 students have viewed full stepbystep solutions from this chapter. Linear Algebra Done Right (Undergraduate Texts in Mathematics) was written by and is associated to the ISBN: 9783319110790. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3. Chapter 5.B: Eigenvectors and UpperTriangular Matrices includes 20 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Block matrix.
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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column space C (A) =
space of all combinations of the columns of A.

Cyclic shift
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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Lucas numbers
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 = MI 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·