- 2.3.1: Determine the transpose of each of the following matrices. Indicate...
- 2.3.2: Each of the following matrices is to be symmetric. Determine the el...
- 2.3.3: If A is 4 X 1, Bis 2 X 3, C is 2 X 4, and Dis 1 X 3, determine the ...
- 2.3.4: Prove the following properties of transpose given in Theorem 2.4. (...
- 2.3.5: Prove the following properties of transpose using the results of Th...
- 2.3.6: LetA be a diagonal matrix. Prove that A = At .
- 2.3.7: LetA be a square matrix. Prove that (An )t = (At)n .
- 2.3.8: Prove that a square matrix A is symmetric if and only if aij = aji ...
- 2.3.9: Let A be a symmetric matrix. Prove that At is symmetric.
- 2.3.10: Prove that the sum of two symmetric matrices of the same size is sy...
- 2.3.11: A square matrixA is said to be antisymmetric if A = -At . (a) Give ...
- 2.3.12: If A is a square matrix prove that (a) A + At is symmetric. (b) A -...
- 2.3.13: Prove that any square matrix A can be decomposed into the sum of a ...
- 2.3.14: a) Prove thatifA is idempotent, thenAt is also idempotent. (b) Prov...
- 2.3.15: Determine the trace of each of the following matrices. [ _ -!J u 1 ...
- 2.3.16: Prove the following properties of trace given in Theorem 2.5. (a) t...
- 2.3.17: Prove the following property of trace using the results of Theorem2...
- 2.3.18: Consider two matrices A and B of the same size. Prove that A= B if ...
- 2.3.19: Prove that the matrix product AB exists if and only if the number o...
- 2.3.20: Prove that AB = On for all n X n matrices B if and only if A= On
- 2.3.21: Compute A + B, AB, and BA for the matrices [ 5 3 - i] B = [-2 + i 5...
- 2.3.22: Compute A + B, AB, and BA for the matrices A= B= [ 4 + i 2 -3i] [2 ...
- 2.3.23: Find the conjugate and conjugate transpose of each of the following...
- 2.3.24: Find the conjugate and conjugate transpose of each of the following...
- 2.3.25: Prove the following four properties of conjugate transpose. (a) (A ...
- 2.3.26: Prove that the diagonal elements of a hermitian matrix are real num...
- 2.3.27: The following matrices describe the pottery contents of various gra...
- 2.3.28: Let G = AA1 and P = A1A, for an arbitrary matrix A. (a) Prove that ...
- 2.3.29: LetA be an arbitrary matrix. What information does the ith diagonal...
- 2.3.30: Derive the result for analyzing the pottery in graves. Let A descri...
- 2.3.31: The model introduced here in archaeology is used in sociology to an...
Solutions for Chapter 2.3: Symmetric Matrices and Seriation in Archaeology
Full solutions for Linear Algebra with Applications | 8th Edition
Tv = Av + Vo = linear transformation plus shift.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
peA) = det(A - AI) has peA) = zero matrix.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).