A note on pseudocongruent matrices

Linear Algebra and its Applications

The pseudo-determinant Det(A) of a square matrix A is defined as the product of the non-zero eigenvalues of A. It is a basis-independent number which is up to a sign the first non-zero entry of the characteristic polynomial of A. We extend here the Cauchy-Binet formula to pseudo-determinants. More specifically, after proving some properties for pseudo-determinants, we show that for any two n times m matrices F,G, the formula Det(F^T G) = sum_P det(F_P) det(G_P) holds, where det(F_P) runs over all k times k minors of A with k=min(rank(F^TG),rank(G F^T)). A consequence is the following Pythagoras theorem: for any self-adjoint matrix A of rank k one has Det^2(A) = sum_P det^2(A_P), where the right hand side sums over the squares over all k times k minors of A.

Proceedings of the American Mathematical Society, 1971

An operator in a Banach space is called generalized pseudo-hermitian if it is a generalized scalar in the sense of Foiaç (i.e., admits a spectral distribution) and has a real spectrum. In this paper this class of operators is characterized by the condition that the real exponential group generated by such an operator has polynomial growth in the uniform operator norm.

Linear Algebra and its Applications, 2021

The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which A * and A † commute, Linear and Multilinear Algebra 14 (1984) 241-256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings.

Journal of Mathematical Analysis and Applications, 2016

We consider the class of non-Hermitian operators represented by infinite tridiagonal matrices, selfadjoint in an indefinite inner product space with one negative square. We approximate them with their finite truncations. Both infinite and truncated matrices have eigenvalues of nonpositive type: either a single one on the real axis or a couple of complex conjugate ones. As a tool to evaluate the reliability of the use of truncations in numerical simulations, we give bounds for the rate of convergence of their eigenvalues of nonpositive type. Numerical examples illustrate our results.

Tamkang Journal of Mathematics, 2015

The analysis on manifolds with singularities is a rapidly developing field of research, with new achievements and compelling challenges. We present here elements of an iterative approach to building up pseudo-differential structures. Those participate in operator algebras on singular manifolds and reflect the properties of parametrices of elliptic operators, including boundary value problems.

2010

The field of quaternions, denoted by H can be represented as an isomorphic four dimensional subspace of R4×4, the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in R4×4 which is also a field and which has in connection with the quaternions many pleasant properties. This field is called field of pseudoquaternions. It exists in R4×4 but not in H. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in R. And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b. Now, the field of quaternions can also be represented as an isomorphic four dimensional subspace of C2×2 over R, the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from R4×4 do not exist. However, there is a subset o...

Methods and Applications of Analysis, 2003

Doklady Mathematics, 2008

Tokyo Journal of Mathematics, 1984