 5.C.1: Suppose T 2 L.V / is diagonalizable. Prove that V D null T range T.
 5.C.2: Prove the converse of the statement in the exercise above or give a...
 5.C.3: Suppose V is finitedimensional and T 2 L.V /. Prove that the follo...
 5.C.4: Give an example to show that the exercise above is false without th...
 5.C.5: Suppose V is a finitedimensional complex vector space and T 2 L.V ...
 5.C.6: Suppose V is finitedimensional, T 2 L.V / has dim V distinct eigen...
 5.C.7: Suppose T 2 L.V / has a diagonal matrix A with respect to some basi...
 5.C.8: Suppose T 2 L.F5/ and dim E.8; T / D 4. Prove that T 2I or T 6Iis i...
 5.C.9: Suppose T 2 L.V / is invertible. Prove that E.; T / D E. 1; T 1/ fo...
 5.C.10: Suppose that V is finitedimensional and T 2 L.V /. Let 1;:::;mdeno...
 5.C.11: Verify the assertion in Example 5.40.
 5.C.12: Suppose R; T 2 L.F3/ each have 2, 6, 7 as eigenvalues. Prove that t...
 5.C.13: Find R; T 2 L.F4/ such that R and T each have 2, 6, 7 as eigenvalue...
 5.C.14: Find T 2 L.C3/ such that 6 and 7 are eigenvalues of T and such that...
 5.C.15: Suppose T 2 L.C3/ is such that 6 and 7 are eigenvalues of T. Furthe...
 5.C.16: The Fibonacci sequence F1; F2;::: is defined byF1 D 1; F2 D 1; and ...
Solutions for Chapter 5.C: Eigenspaces and Diagonal Matrices
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics)  3rd Edition
ISBN: 9783319110790
Solutions for Chapter 5.C: Eigenspaces and Diagonal Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 5.C: Eigenspaces and Diagonal Matrices have been answered, more than 6605 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 textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3. Chapter 5.C: Eigenspaces and Diagonal Matrices includes 16 full stepbystep solutions.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt 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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.