- 61.61.1: Allfigures referred to in 61.1-61.5 are assumed to be extended to f...
- 61.61.2: Allfigures referred to in 61.1-61.5 are assumed to be extended to f...
- 61.61.3: Allfigures referred to in 61.1-61.5 are assumed to be extended to f...
- 61.61.4: Allfigures referred to in 61.1-61.5 are assumed to be extended to f...
- 61.61.5: Allfigures referred to in 61.1-61.5 are assumed to be extended to f...
- 61.61.6: Verify that the rotations in the 32 crystallographic point groups a...
- 61.61.7: Prove that the only values of n for which the plane can be filled w...
Solutions for Chapter 61: ON CRYSTALLOGRAPHIC GROUPS
Full solutions for Modern Algebra: An Introduction | 6th Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Invert A by row operations on [A I] to reach [I A-I].
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.