A matrix is called strictly sign-regular of order k (denoted by S S R k ) if all its k 脳 k minors are non-zero and have the same sign. For example, totally positive matrices, i.e., matrices with all minors positive, are S S R k for all k . Another important subclass are ...
A matrix is said to be a signature matrix if J is diagonal and its diagonal entries are . If J is a signature matrix, a nonsingular matrix is said to be a J-orthogonal matrix if . Let be the set of all , J-orthogonal matrices. In this paper some further interesting properties of ...
Since the determinant is zero, a singular matrix is non-invertible, which does not have an inverse. What is an example of a singular matrix? A square matrix whose determinant is zero is an example of a singular matrix. This could be a 2x2 matrix with entries a, b, c, d, where a=...
We determine whether a matrix is a singular matrix or a non-singular matrix depending on its determinant. The determinant of a matrix 'A' is denoted by 'det A' or '|A|'. If the determinant of a matrix is 0, then it is said to be a singular matrix. Why do we need to have a ...
A row matrix is a type of a matrix that has only one row. The total number of columns in a row matrix is the total number of elements that make up the single row. The row matrix is not a square matrix as the number of rows is not equal to the number of columns. Thus, we ...
If A is nonsingular, then S is also nonsingular. Theorem 3: If A is an n × n matrix of rank m, then A is TP iff every minor of A formed from any columns β 1,…,β p satisfying ∑ i=2 p |β iβ i11| nm is nonnegative. Theorem 4: If A is a nonsingular lower ...
An invertible matrix in linear algebra (also called non-singular or non-degenerate), is the n-by-n square matrix satisfying the requisite condition for the inverse of a matrix to exist, i.e., the product of the matrix, and its inverse is the identity matrix. What are the Properties of...
摘要: The paper deals with a stable homotopy classification of continuous families of invertible matrix one-dimensional singular integral operators with piecewise continuous coefficients in the space L p n (Γ), where Γ is a simple closed Lyapunov curve....
For a Hermitian matrix H with nonsingular principal submatrix A , it is shown that the eigenvalues of the Moore-Penrose inverse of the Schur complement ( H / A ) of A in H interlace the eigenvalues of the Moore-Penrose inverse of H . Moreover, if H is semidefinite, it is shown that...
Williams [13], with new proofs of many results of Dixmier. In this paper we shall pursue a few problems on operator ranges, most related with additive properties; for instance, describe for which A,B∈L(H) it holds R(A+B)=R(A)+R(B), where L(H) denotes the algebra of bounded ...