 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
ISBN: 9781449679545
Solutions for Chapter 2.3: Symmetric Matrices and Seriation in Archaeology
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 31 problems in chapter 2.3: Symmetric Matrices and Seriation in Archaeology have been answered, more than 8759 students have viewed full stepbystep solutions from this chapter. Chapter 2.3: Symmetric Matrices and Seriation in Archaeology includes 31 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545.

Affine transformation
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.

CayleyHamilton Theorem.
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 edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
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).

Network.
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.

Normal matrix.
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.

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 ().

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 = UI.
Orthonormal columns (complex analog of Q).