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On the complexity of matrix product
Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, 2002Summary: Our main result is a lower bound of \(\Omega(m^2 \log m)\) for the size of any arithmetic circuit for the product of two matrices, over the real or complex numbers, as long as the circuit does not use products with field elements of absolute value larger than 1 (where \(m\times m\) is the size of each matrix).
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Displacements of matrix products
1995For fixed matrices M and N, either of the linear transformations A ↦ A-MAN or or A ↦ MA-AN is called a displacement of the matrix A. Displacement can greatly reduce the rank of structured matrices, such as circulant, Vandermonde, Toeplitz and Hankel matrices. This rank reduction has been widely used for inverting structured matrices.
Quyen L. Nguyen, David H. Wood
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Addition requirements for matrix and transposed matrix products
Journal of Algorithms, 1988The authors study the complexity, specifically the number of additions, of linear algorithms which compute a set of linear forms defined by a given \(s\times t\) matrix M. The developed linear algorithms are defined as labeled directed acyclic graphs.
Michael Kaminski +2 more
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Sparse Matrix-Matrix Products Executed Through Coloring
SIAM Journal on Matrix Analysis and Applications, 2015Summary: Sparse matrix-matrix products appear in multigrid solvers among other applications. Some implementations of these products require the inner product of two sparse vectors. In this paper, we propose a new algorithm for computing sparse matrix-matrix products by exploiting their nonzero structure through the process of graph coloring.
Michael McCourt +2 more
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Inequalities for the trace of matrix product
IEEE Transactions on Automatic Control, 1994To obtain estimates of solutions of Lyapunov and Riccati equations which frequently occur in the stability analysis and optimal control design in linear control theory, many researchers have attempted to determine upper and lower bounds for the product of two matrices in terms of the trace of one matrix and the eigenvalues of the other.
Yuguang Fang +2 more
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1987
Turning back to the correlation matrix Σ = (σαβ) associated, in the previous chapter, with the primitive and aperiodic substitution ζ of length q, we shall prove that Σ is the weak-star limit point of a product of matrices whose entries are trigonometric polynomials, in a way similar to the case of generalized Riesz products.
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Turning back to the correlation matrix Σ = (σαβ) associated, in the previous chapter, with the primitive and aperiodic substitution ζ of length q, we shall prove that Σ is the weak-star limit point of a product of matrices whose entries are trigonometric polynomials, in a way similar to the case of generalized Riesz products.
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Eigenvalue Inequalities for Matrix Product
IEEE Transactions on Automatic Control, 2006We present a family of eigenvalue inequalities for the product of a Hermitian matrix and a positive-semidefinite matrix.
Fuzhen Zhang, Qingling Zhang 0001
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On the trace bound of a matrix product
IEEE Transactions on Automatic Control, 1996With the basic equations of state-space control theory, the matrix Riccati and Lyapunov equations are connected to a priori estimates of their solutions. Fang-Loparo-Feng in 1994 improved such estimates; the authors continue this work, giving better lower and upper bounds of the product of two matrices. Introducing a sparse matrix to diagonalize one of
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Quantifying matrix product state
Quantum Information Processing, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Amandeep Singh Bhatia, Ajay Kumar 0003
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Equivalence of a matrix product to the Kronecker product.
2000The authors give an interesting proof of the permutation equivalence of the Tracy-Singh product for partitioned matrices[\textit{D. S. Tracy} and \textit{R. P. Singh}, Stat. Neerl. 26, 143--157 (1972; Zbl 0267.15009); \textit{S. Liu}, Linear Algebra Appl. 289, No. 1-3, 267--277 (1999; Zbl 0937.15015)] to the Kronecker product.
Wei, Yimin, Zhang, Fuzhen
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