 6.2.1: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.2: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.3: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.4: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.5: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.6: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.7: Questions 17 are about t~e eigenvalue and eigenvector matrices A a...
 6.2.8: Questions 810 are about Fibonacci and Gibonacci numbers.
 6.2.9: Questions 810 are about Fibonacci and Gibonacci numbers.
 6.2.10: Questions 810 are about Fibonacci and Gibonacci numbers.
 6.2.11: True or false: If the eigenvalues of A are 2, 2, 5 then the matrix ...
 6.2.12: True or false: If the only eigenvectors of A are mUltiples of (1, 4...
 6.2.13: Complete these matrices so that det A = 25. Then check that A = 5 i...
 6.2.14: The matrix A = [~~] is not diagonalizable because the rank of A  3...
 6.2.15: Questions 1519 are about powers of matrices.
 6.2.16: Questions 1519 are about powers of matrices.
 6.2.17: Questions 1519 are about powers of matrices.
 6.2.18: Questions 1519 are about powers of matrices.
 6.2.19: Questions 1519 are about powers of matrices.
 6.2.20: Suppose A = SASl. Take determinants to prove detA = detA = AIA2 An...
 6.2.21: Show that trace S T = trace T S, by adding the diagonal entries of ...
 6.2.22: AB  BA = I is impossible since the left side has trace = __ . But ...
 6.2.23: If A = SAS 1 , diagonalize the block matrix B = [~J]' Find its eig...
 6.2.24: Consider all 4 by 4 matrices A that are diagonalized by the same fi...
 6.2.25: Suppose A2 = A. On the left side A multiplies each column of A. Whi...
 6.2.26: (Recommended) Suppose Ax = AX. If A = 0 then X is in the nUllspace....
 6.2.27: The eigenvalues of A are 1 and 9, and the eigenvalues of Bare 1 an...
 6.2.28: (Heisenberg's Uncertainty Principle) AB  BA = I can happen for inf...
 6.2.29: If A and B have the same A'S with the same independent eigenvectors...
 6.2.30: Suppose the same S diagonalizes both A and B. They have the same ei...
 6.2.31: (a) If A = [g ~] then the determinant of A  AI is (A  a)(A  d). ...
 6.2.32: Substitute A = SAS1 into the product (A  )"1 I) (A  A2I) (A  An...
 6.2.33: Find the eigenvalues and eigenvectors and the kth power of A. For t...
 6.2.34: If A = [ij ~J and AB = BA, show that B = [~~J is also a diagonal ma...
 6.2.35: The powers Ak approach zero if all lAd < 1 and they blow up if any ...
 6.2.36: The nth power of rotation through () is rotation through n(): An = ...
 6.2.37: The transpose of A = SAS1 is AT = (Sl)T AST. The eigenvectors in ...
 6.2.38: The inverse of A = eye(n) + ones(n) is AI = eye(n) + C * ones(n). ...
Solutions for Chapter 6.2: Diagonalizing a Matrix
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 6.2: Diagonalizing a Matrix
Get Full SolutionsThis textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Chapter 6.2: Diagonalizing a Matrix includes 38 full stepbystep solutions. Since 38 problems in chapter 6.2: Diagonalizing a Matrix have been answered, more than 12867 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714.

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

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).