lemmastandard

Moore determinant and cofactor properties

Shows Moore cofactor

C(A)=(\det A)A^{-1}

, nonnegative, degree

m-1

linear-algebra facts for cofactor and Dirichlet estimates

Lemma·lem:quaternionic-matrices·§3.2 The Moore cofactor

Statement (paper)

The Moore determinant is a homogeneous polynomial of degree

m

. Its cofactor

C(A)

is the Hermitian matrix defined by

C(A):B=\left.\frac{d}{dt}\det(A+tB)\right|_{t=0}.

It is polynomial of degree

m-1

, and

C(A)=(\det A) A^{-1}\quad (A>0).

A\ge0

, then

C(A)\ge0

; if also

A

is singular,

C(A)

kills the image of

A

. On positive definite matrices

\log\det

is concave and increasing in the matrix order. For a unit vector

v\in\HH^m

let

A|_{v^\perp}

denote the compression of the form to its quaternionic orthogonal complement. For every Hermitian

A

and real

t

v^*C(A)v=\det(A|_{v^\perp}),\qquad \det(A+t vv^*)=\det A+t\,v^*C(A)v.

The determinant on the zero-dimensional space is one. Finally, for a Hermitian

M=\bigl(\begin{smallmatrix}\ell&c^*\\ c&D\end{smallmatrix}\bigr)

with

D>0

, put

\sigma=\ell-c^*D^{-1}c

and let

v_0

be the first coordinate vector. Then

\det M=(\det D)\sigma,\qquad M-\sigma v_0v_0^*\ge0 .

In plain English

For quaternionic Hermitian

m\times m

matrices, the Moore determinant is homogeneous of degree

m

, and its cofactor

C(A)

, defined as the derivative of

\det(A+tB)

in direction

B

, is a degree

m-1

polynomial equal to

(\det A)A^{-1}

for

A>0

. Moreover

C(A)\ge0

for

A\ge0

\log\det

is concave and monotone on positive definite matrices,

v^*C(A)v=\det(A|_{v^\perp})

for unit

v

, and a Schur complement formula gives

\det M=(\det D)\sigma

with

M-\sigma v_0v_0^*\ge0

Why it's here

This lemma collects the standard linear-algebra facts about quaternionic determinants that the analytic arguments need: the cofactor plays the role of the linearized determinant, and the compression and Schur formulas let one compute or bound it along a direction. Later results such as the cofactor divergence identity, the energy balance, the Dirichlet problem estimates and the double normal derivative bound all rely on these facts.

Classified standard:

Standard linear algebra of quaternionic Hermitian matrices via the Moore determinant: cofactor formula, log-concavity, compression and Schur complement identities (Moore 1922; see Aslaksen 1996 'Quaternionic determinants', and Alesker's work on quaternionic Monge–Ampère).

Uses

Nothing else in the paper

Used by

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Sources

PaperOpen

Section: §3.2 The Moore cofactor
Page: —
Label: lem:quaternionic-matrices
Line: sections/calculus.tex:99

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