1. Let A be an n× n matrix with complex coefficients. Define trA to be the sum of the diagonal elements. Show that trA is invariant under conjugation, i.e., trA = trPAP for all invertible n× n matrices P. Proof. Let P be an invertible matrix. Let ~pk be the k-th row of P, ~qj the j-th column of P, and ~ ai the i-th column of A. The k-th row of the matrix PA is 〈~pk · ~ a1, . . . ,~pk · ~ an〉. S...

A matrix of discrimination measures (discrimination probabilities, numerical estimates of dissimilarity, etc.) satisfies Regular Minimality (RM) if every row and every column of the matrix contains a single minimal entry, and an entry minimal in its row is minimal in its column. We derive a formula for the proportion of RM-compliant matrices among all square matrices of a given size and with no...

We show that a cubic graph G with girth g(G) ≥ 5 has a Hamiltonian Circuit if and only if the matrix A+ I can be row permuted such that each column has at most 2 blocks of consecutive 1’s, where A is the adjacency matrix of G, I is the unit matrix, and a block can be consecutive in circular sense, i.e., the first row and the last row are viewed as adjacent rows. Then, based on this necessary an...

Barycenter The mean of the observations from a given category (also called center of gravity, center of mass, mean vector, or centroid) Confidence interval An interval encompassing a given proportion (e.g., 95%) of an estimate of a parameter (e.g., a mean) Discriminant analysis A technique whose goal is to assign observations to some predetermined categories Discriminant factor scores A linear ...

We review the polynomial matrix compensator equation XlDr + YlNr = Dk (COMP), e.g. (Callier and Desoer, 1982, Kučera, 1979; 1991), where (a) the right-coprime polynomial matrix pair (Nr,Dr) is given by the strictly proper rational plant right matrix-fraction P = NrD−1 r , (b) Dk is a given nonsingular stable closed-loop characteristic polynomial matrix, and (c) (Xl, Yl) is a polynomial matrix s...

The problem of matrix inversion is central to many applications of Numerical Linear Algebra. When the matrix to invert is dense, little can be done to avoid the costly O(n) process of Gaussian Elimination. When the matrix is symmetric, one can use the Cholesky Factorization to reduce the work of inversion (still O(n), but with a smaller coefficient). When the matrix is both sparse and symmetric...

In this paper, the eigenvalues of row-inverted 2 × 2 Sylvester Hadamard matrices are derived. Especially when the sign of a single row or two rows of a 2×2 Sylvester Hadamard matrix are inverted, its eigenvalues are completely evaluated. As an example, we completely list all the eigenvalues of 256 different row-inverted Sylvester Hadamard matrices of size 8. Mathematics Subject Classification (...

An equitable vertex coloring for the splitting of block circulant graphs is investigated. The comprises matrices, where each itself a matrix. These blocks in row are cyclically shifted one place to right from those previous row. We approached such matrix representation and derived their independent sets using neighbourhoods vertex. This classification makes process be simpler most cases. In thi...

In this article, we present several new permutations for I-matrices making these more suitable for incomplete LDU-factorization preconditioners used in solving linear systems by iterative methods. A general matrix can be transformed by row permutation as well as row and columns scaling into an I-matrix, i.e. a matrix having elements of modulus 1 on the diagonal and elements of modulus of no mor...

We determine analytically the condition number of the PageRank problem. Specifically, we prove the following statement: “Let P be an n × n row-stochastic matrix whose diagonal elements Pii = 0. Let c be a real number such that 0 ≤ c < 1. Let E be the n × n rank-one row-stochastic matrix E = ev , where e is the n-vector whose elements are all ei = 1, and v is an n-vector that represents a probab...