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Matrices and Determinants
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
Some properties of unitary matrices are: If a matrix R is unitary, then it is nonsingular, and R−1=R†.The rows of a unitary matrix form an orthonormal set of vectors. Similarly, the columns of a unitary matrix form an orthonormal set of vectors.The product of two unitary matrices is a unitary matrix.If a matrix R is unitary, then |det (R) | = 1.All eigenvalues of a unitary matrix have a unit modulus (magnitude).Let R be a unitary matrix. If matrices A and B are related to each other via a unitary transformation, that is if A=R†BR, then the matrices A and B have the same eigenvalues.
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
A complex matrix U of order n is called unitary if UU⋆=In (Recall that U⋆ is the conjugate transpose of U). Again, this means that U⋆U=In. Also a real unitary matrix is simply an orthogonal matrix. The unit matrix is both orthogonal as well as unitary. For example, the matrix U=(1/5)(−1+2i−4−2i2−4i−2−i) is unitary.
O
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
orthogonal code division multiple access scheme a code division multiple access scheme (CDMA), i.e., a spread spectrum system with multiple users, where the PN sequences utilized by the different users are orthogonal sequences. In such a system, the multiple access interference is zero. The orthogonal sequences can be obtained from the rows of a Hadamard matrix, for example. orthogonal CDMA See orthogonal code division multiple access scheme. orthogonal filter bank a biorthogonal filter bank whose polyphase transfer functions of the analysis and synthesis filters satisfy F(z) = H T (z -1 ). orthogonal frequency division multiplex (OFDM) a frequency division multiplex scheme where the subcarriers for each of the frequency divided bands (subbands) are orthononal to each other. This allows for spectral overlap between successive subbands. orthogonal hopping sequences a set of frequency hopping sequences (hopping patterns) for which no two sequences have the same value for a given position in the sequence. The set of sequences can be represented as the rows of a matrix. In such a case for a given column of the matrix, the elements are distinct. orthogonal matrix a matrix A whose inverse is A T ; that is, A T A = I , where I is the identity matrix. More generally, A is a unitary matrix if its inverse is A H (that is, the complex-conjugate transpose); consequently a real unitary matrix is an orthogonal matrix. Although not recommended, the phrase "orthogonal matrix" is sometimes used to refer to a unitary matrix.
Optimal blind colour image watermarking based on adaptive chaotic grasshopper optimization algorithm
Published in The Imaging Science Journal, 2022
In linear algebra, one of the important matrix decomposition tools is Schur decomposition which is the intermediary step in SVD. The computations required for Schur decomposition or Schur triangulation are three times lower than SVD. The word Schur is termed after the popular Russian mathematician Issai Schur. The factorization of a real matrix is given as where is a unitary matrix (i.e. ) and its conjugate transpose is . Here denotes the upper triangular matrix whose diagonal entries comprise of the eigenvalues of .
Digital video watermarking algorithm based on asymmetric causal spatio-temporal filter
Published in Systems Science & Control Engineering, 2018
Yanliang Ge, Hongbo Bi, Di Lu, Xiaoxue Kong, Zheng Zhang, Ning Li
The new matrix F is decomposed using Singular Value Decomposition (SVD). where , is a unitary matrix, and , and , is an identity matrix, represents the conjugate transpose. is a non-negative diagonal matrix of diagonal elements. where is the rank of , is named singular value (SV), while is named eigenvalue.