On Some Properties of Convex Matrix Sets Characterized byP-Matrices and BlockP-Matrices

Abstract

In this paper, we consider convex sets of real matrices and establish criteria characterizing these sets with respect to certain matrix properties of their elements. In particular, we deal with convex sets of P-matrices, block P-matrices and M-matrices, nonsingular and full rank matrices, as well as stable and Schur stable matrices. Our results are essentially based on the notion of a block P-matrix and extend and generalize some recently published results on this topic.

A practical test for assessing the reachability of discrete-time Takagi–Sugeno fuzzy systems

“…Real P-matrices are well known in matrix theory because they play an important role in many applications [40]. In [41], it was shown that the P-property of a single matrix is equivalent to the nonsingularity of all matrices in a certain convex matrix set.…”

“…A matrix A ∈ R n×n is called a P-matrix if all its principal minors are positive [40]. On the other hand, a matrix A ∈ R n×n is a block P-matrix with respect to a partition N(λ ) of N = {1, .…”

“…is the principal submatrix of T with row and column indices in N i [40]. A P-matrix is also block P-matrix with respect to any partition [40].…”

A necessary and sufficient condition for total observability of discrete-time linear time-varying systems

“…Real P-matrices are well known in matrix theory because they play an important role in many applications (Elsner et al, 2002). In Johnson and Tsatsomeros (1995), it was shown that the P-property of a single matrix is equivalent to the nonsingularity of all matrices in a certain convex matrix set.…”

“…The matrix X ∈ R n×n is called a P-matrix if all its principal minors are positive (Elsner et al, 2002). On the other hand, a matrix X ∈ R n×n is a block P-matrix with respect to a partition .…”