Use of Jordan forms for convection-pressure split Euler solvers

In many basic linear algebra texts it is shown that various classes of square matrices (normal, positive, invertible) possess square roots. In this note we characterize those n × n matrices with complex entries which possess at least one square root without any restriction on the class of root or matrix involved. We then use this characterization to obtain asymptotic estimates for the relative profusion of such matrices.In Section 1 we characterize those n × n matrices with entries in C (or any algebraically complete field) which have square roots over C. This characterization is in terms of similarity classes. In Section 2 we give asymptotic estimates for the number of Jordan forms of nilpotent n × n matrices which are squares. Section 3 is given over to numerical results concerning the actual and asymptotic frequency of such forms.

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q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2362 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Martha Yip

Keyword(s):

Finite Field ◽

Weighted Sum ◽

Canonical Forms ◽

Combinatorial Formula ◽

Jordan Type ◽

Rook Placements ◽

Nilpotent Matrices ◽

International Audience ◽

Jordan Forms

International audience The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$ F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements.

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Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices

The Electronic Journal of Combinatorics ◽

10.37236/6888 ◽

2018 ◽

Vol 25 (1) ◽

Author(s):

Martha Yip

Keyword(s):

Finite Field ◽

Weighted Sum ◽

Young Tableaux ◽

Canonical Forms ◽

Combinatorial Formula ◽

Rook Placements ◽

Nilpotent Matrices ◽

Jordan Forms ◽

Standard Young Tableaux ◽

Young’S Lattice

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by partitions $\lambda \vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for $F_\lambda(q)$.

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