 6.3.1: In each of the following, factor the matrix A into a product XDX1, ...
 6.3.2: For each of the matrices in Exercise 1, use the XDX1 factorization ...
 6.3.3: For each of the nonsingular matrices in Exercise 1, use the XDX1 fa...
 6.3.4: For each of the following, find a matrix B such that B2 = A: (a) A ...
 6.3.5: Let A be a nondefective nn matrix with diagonalizing matrix X. Show...
 6.3.6: Let A be a diagonalizable matrix whose eigenvalues are all either 1...
 6.3.7: Show that any 3 3 matrix of the form a 1 0 0 a 1 0 0 b is defective.
 6.3.8: For each of the following, find all possible values of the scalar t...
 6.3.9: Let A be a 44 matrix and let be an eigenvalue of multiplicity 3. If...
 6.3.10: Let A be an n n matrix with positive real eigenvalues 1 > 2 > > n. ...
 6.3.11: Let A be a n n matrix with real entries and let 1 = a + bi (where a...
 6.3.12: Let A be an n n matrix with an eigenvalue of multiplicity n. Show t...
 6.3.13: Show that a nonzero nilpotent matrix is defective. 1
 6.3.14: Let A be a diagonalizable matrix and let X be the diagonalizing mat...
 6.3.15: It follows from Exercise 14 that, for a diagonalizable matrix, the ...
 6.3.16: Let A be an n n matrix and let be an eigenvalue of A whose eigenspa...
 6.3.17: Let x, y be nonzero vectors in Rn, n 2, and let A = xyT . Show that...
 6.3.18: Let A be a diagonalizable n n matrix. Prove that if B is any matrix...
 6.3.19: Show that if A and B are two n n matrices with the same diagonalizi...
 6.3.20: Let T be an upper triangular matrix with distinct diagonal entries ...
 6.3.21: Each year, employees at a company are given the option of donating ...
 6.3.22: The city of Mawtookit maintains a constant population of 300,000 pe...
 6.3.23: Let A = 1 2 1 3 1 5 1 4 1 3 2 5 1 4 1 3 2 5 be a transition matrix ...
 6.3.24: Consider a Web network consisting of only four sites that are linke...
 6.3.25: Let A be an n n stochastic matrix and let e be the vector in Rn who...
 6.3.26: The transition matrix in Example 5 has the property that both its r...
 6.3.27: Let A be the PageRank transition matrix and let xk be a vector in t...
 6.3.28: Use the definition of the matrix exponential to compute eA for each...
 6.3.29: Compute eA for each of the following matrices: (a) A = 2 1 6 3 (b) ...
 6.3.30: In each of the following, solve the initial value problem Y_ = AY, ...
 6.3.31: Let be an eigenvalue of an n n matrix A and let x be an eigenvector...
 6.3.32: Show that eA is nonsingular for any diagonalizable matrix A. 3
 6.3.33: Let A be a diagonalizable matrix with characteristic polynomial p()...
Solutions for Chapter 6.3: Diagonalization
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 6.3: Diagonalization
Get Full SolutionsSince 33 problems in chapter 6.3: Diagonalization have been answered, more than 6753 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Chapter 6.3: Diagonalization includes 33 full stepbystep solutions.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.