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Linear Algebra and Matrices
Published in William F. Ames, George Cain, Y.L. Tong, W. Glenn Steele, Hugh W. Coleman, Richard L. Kautz, Dan M. Frangopol, Paul Norton, Mathematics for Mechanical Engineers, 2022
The transpose of a matrix A is the matrix that results from interchanging the rows and columns of A. It is usually denoted by AT. A matrix A such that A = AT is said to be symmetric. The conjugate transpose of A is the matrix that results from replacing each element of AT by its complex conjugate, and is usually denoted by AH. A matrix such that A = AH is said to be Hermitian.
Complex numbers
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. Hence the complex conjugate of a+jb is a−jb. The product of a complex number and its complex conjugate is always a real number.
Complex numbers
Published in John Bird, Bird's Engineering Mathematics, 2021
The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. Hence the complex conjugate of a+jb is a−jb. The product of a complex number and its complex conjugate is always a real number.
Performance limitations for discrete-time systems in the presence of multi-sinusoidal disturbance
Published in International Journal of Control, 2021
Notations: The notation used throughout this paper is fairly standard. denotes the imaginary number. denotes the -transform of discrete-time series . , denote the expectation operator and variance operator of a discrete-time series, respectively. , denote the complex valued plane and the set of real number. For any complex number , denotes the complex conjugate. For a vector X, denotes the ith element, and for a matrix M, denotes the element in ith row and jth column. Let the open unit circle be denoted by , the close unit circle denoted by , and the unit circle denoted by . Let the complement of be denoted by . is the standard frequency domain Lebesgue space. and are subspaces of containing functions that are analytic in and , respectively. It is well known that and constitute orthogonal complements in . denotes a space of scalar-valued functions that are essentially bounded on . is a subspace in with functions that are bounded on and analytic in . The real rational subspace of is denoted by which consists of all proper and real rational stable transfer functions. For any transfer function , we denote its -norm by . Res denotes the residue of at point .