Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

On determinantal representation for the generalized inverse A (2)T,S and its applications

Abstract: In this paper, first we establish a determinantal representation for the group inverse Ag of a square matrix A. Based on this, a determinantal representation for the generalized inverse A is presented. As an application, we give a determinantal formula for the unique solution of the general restricted linear system: Ax=b(x ∈ T, b ∈ AT and dim(AT)=dim(T)), which reduces to the common Cramer rule if A is non-singular. These results extend our earlier work. Copyright © 2006 John Wiley & Sons, Ltd.

On the Theory of Quadratic Forms

Abstract: In this note we give an extention of the analytic theory of quadratic forms of C. L. Siegel'. We use the well-known matrix notation, if the contrary is not expressly mentioned we suppose the elements of all matrices to be rational integers. Let A(S, T; P, v) denote the number of solutions X of the Diophantic matrix equation X'SX = T, which satisfy the congruence X _ P (mod v). We suppose henceforth the matrices S and T to be symmetric positive matrices of degree m and n respectively, and P = p(mn) to be a primitive matrix modulo v. A matrix is called primitive mod v if it can be completed to a square matrix, the determinant of which is coprime with v. The number v is a positive integer. All matrices U'S U, where U is a unimodular matrix satisfying U = E(mod v) are said to belong to the principal class $3 z(S, v) of S. All matrices S,, such that a matrix U, with (I U 1, q) = 1 can be found satisfying U'SU S, (mod q); U =_ E (mod v) are said to belong to the principal class q(S, v) of S modulo q. If a matrix S, belongs to $1, q(S, v) for all positive integers q, we say that the matrix S, belongs to the principal genus $5(S, v) of S.2 Let Sk run through a complete system of principal class representatives of the principal genus of S. Then we define

Low-footprint adaptation and personalization for a deep neural network

Abstract: The adaptation and personalization of a deep neural network (DNN) model for automatic speech recognition is provided. An utterance which includes speech features for one or more speakers may be received in ASR tasks such as voice search or short message dictation. A decomposition approach may then be applied to an original matrix in the DNN model. In response to applying the decomposition approach, the original matrix may be converted into multiple new matrices which are smaller than the original matrix. A square matrix may then be added to the new matrices. Speaker-specific parameters may then be stored in the square matrix. The DNN model may then be adapted by updating the square matrix. This process may be applied to all of a number of original matrices in the DNN model. The adapted DNN model may include a reduced number of parameters than those received in the original DNN model.

Factorization of the transfer matrix for symmetrical optical systems

Abstract: In the paraxial approximation a symmetrical optical system may be represented by a 2 × 2 matrix. It has been the custom to describe each optical element by a transfer matrix representing propagation between the principal planes or through an interface for thin elements. If the focal-plane representation is used instead, any focusing element or combination of elements is represented by the same antidiagonal matrix whose nonzero elements are the focal lengths: The matrix represents propagation between the focal planes. For propagation between any two arbitrary planes, the system transfer matrix can be decomposed into the product of two upper triangular matrices and an antidiagonal matrix. This decomposition yields the above-mentioned focal-plane matrix, and the two upper triangular matrices represent propagation between the input and the output planes and the focal planes. Because the matrix decomposition directly yields the parameters of interest, the analysis and the synthesis of optical systems are simpler to carry out. Examples are given for lenses, diopters, mirrors, periodic sequences, resonators, lenslike media, and phase-conjugate mirror systems.