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Matrices
Published in James R. Kirkwood, Bessie H. Kirkwood, Elementary Linear Algebra, 2017
James R. Kirkwood, Bessie H. Kirkwood
A zero matrix is a matrix for which all of the entries are 0. We denote the zero matrix of size m × n by 0˜m×n. If the dimensions are obvious, we will simply use 0˜. We have the desired property that if A is an m × n matrix, then A+0˜m×n=A. Matrix multiplication presents some challenges that do not occur with matrix addition. The most prominent of these is that matrix multiplication is normally not commutative even if the dimensions allow for multiplication. That is, it is most often the case that for matrices A and BAB≠BA. We expand on this. If A is an m × k matrix and B is a k × n matrix then AB is an m × n matrix, but BA is not defined unless m = n. If m = n then AB is an m × m matrix, but BA is a k × k matrix.
Matrices
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
A zero matrix is a matrix for which all of the entries are 0. We denote the zero matrix of size m × n by 0˜m×n.
Matrices over Integral Domains
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
For A ∈ Dm×n, the rank of A is the maximal size of the nonvanishing minor. (The rank of zero matrix is 0.)
Finite-time event-triggered filtering of the discrete-time switched linear systems with time delay
Published in International Journal of General Systems, 2022
Ziyi He, Baowei Wu, Lili Liu, Qian Bai
Notations: Let be the space of n dimensional vectors, and be the set of all dimensional real matrices. is the natural number set, and , . For a matrix P, P>0 signifies that P is symmetric positive definite matrix. , and express the inverse, transpose and inverse transpose of P, and * denotes the symmetrical items in a symmetric matrix. and is respectively the minimum and maximum eigenvalues of P. I and 0 are identity matrix and zero matrix, respectively. Matrices have the compatible dimensions unless special declarations are made. is the space of square-summable sequences over . is the Delta function. Throughout the whole study, , .
H∞ fault detection for 2-D T–S discrete stochastic fuzzy systems
Published in International Journal of Systems Science, 2020
Xiaofeng Li, Lizhen Li, Shuxia Ye
Notation: The superscript T denotes matrix transposition. In symmetric block matrices or long matrix expressions, we use an asterisk () to represent a term induced by symmetry and diag denotes a block-diagonal matrix. The notation P>0 means that P is real symmetric and positive definite, I and 0 represent the identity matrix and zero matrix, respectively, refers to the Euclidean vector norm. and denote the minimum and the maximum eigenvalues of a real symmetric matrix, respectively. In addition, and , respectively, mean expectation of x and expectation of x conditional on y. A norm of a 2-D signal is given by We say belongs to , if .
Set-membership filtering for piecewise linear systems with censored measurements under Round-Robin protocol
Published in International Journal of Systems Science, 2020
Xuerong Li, Fei Han, Nan Hou, Hongli Dong, Hongjian Liu
Unless otherwise stated, the symbols used here are quite standard. and denote the n dimensional Euclidean space and the set of all real matrices, respectively. The set of nonnegative integers is denoted by . The notation (P>0) means that P is a real, symmetric and positive semi-definite (positive definite) matrix. 0 and I denote the zero matrix and the identity matrix, respectively. means a column which is produced by the items in the set sequentially. For a matrix A, and denote its transpose and inverse, respectively. For a vector a, . The shorthand refers to a block-diagonal matrix. In symmetric block matrices, the symbol represents an ellipsis for terms induced by symmetry. denotes the trace of matrix M. Matrices that do not specify the size should be compatible with matrix operations.