- 7.2.1: If A = 1 2 0 3 2 1 2 1 3 and B = 4 2 3 1 5 0 6 1 2 , find a. 2A + B...
- 7.2.2: If A = _1 + i 1 + 2i 3 + 2i 2 i _ and B = _i 3 2 2i _, find a. A 2B...
- 7.2.3: If A = 2 1 2 1 0 3 2 1 1 and B = 1 2 3 3 1 1 2 1 0 , find a. AT b. ...
- 7.2.4: If A = _3 2i 1 + i 2 i 2 + 3i _, find a. AT b. A c. A
- 7.2.5: If A = 1 2 0 3 2 1 2 0 3 , B = 2 1 1 2 3 3 1 0 2 , and C = 2 1 0 1 ...
- 7.2.6: Prove each of the following laws of matrix algebra: a. A + B = B + ...
- 7.2.7: If x = 2 3i 1 i and y = 1 + i 2 3 i , find a. xT y b. yT y c. (x, y...
- 7.2.8: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.9: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.10: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.11: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.12: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.13: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.14: In each of 8 through 14, if the given matrix is nonsingular, find i...
- 7.2.15: If A is a square matrix, and if there are two matrices B and C such...
- 7.2.16: If A(t) = et 2et e2t 2et et e2t et 3et 2e2t and B(t) = 2et et 3e2t ...
- 7.2.17: In each of 17 and 18, verify that the given vector satisfies the gi...
- 7.2.18: In each of 17 and 18, verify that the given vector satisfies the gi...
- 7.2.19: In each of 19 and 20, verify that the given matrix satisfies the gi...
- 7.2.20: In each of 19 and 20, verify that the given matrix satisfies the gi...
Solutions for Chapter 7.2: Matrices
Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition
peA) = det(A - AI) has peA) = zero matrix.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
cond(A) = c(A) = IIAIlIIA-III = 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.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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 S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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 .
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).