X-matrices

Abstract

We evidence a family $\mathcal{X}$ of square matrices over a field $\mathbb{K}$, whose elements will be called X-matrices. We show that this family is shape invariant under multiplication as well as transposition. We show that $\mathcal{X}$ is a (in general non-commutative) subring of $GL(n,\mathbb{K})$. Moreover, we analyse the condition for a matrix $A \in \mathcal{X}$ to be invertible in $\mathcal{X}$. We also show that, if one adds a symmetry condition called here bi-symmetry, then the set $\mathcal{X}^b$ of bi-symmetric X-matrices is a commutative subring of $\mathcal{X}$. We propose results for eigenvalue inclusion, showing that for X-matrices eigenvalues lie exactly on the boundary of Cassini ovals. It is shown that any monic polynomial on $ \mathbb{R} $ can be associated with a companion matrix in $ \mathcal{X} $.

Figures (1)

• Figure 1: Eigenvalue Inclusion via Cassini ovals for the X matrix $A$, its symmetric product $A^T A$, multiplied by the anti-diagonal, and in X companion form.

Theorems & Definitions (41)

• Definition 1
• Example 2
• Lemma 3
• proof
• Corollary 4
• Corollary 5
• proof
• Definition 6
• Example 7
• Remark 8