16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.4.16

Bilinear Forms

Mathematics — Linear Algebra · symplectic-group-conditions

Problem

Let

n

be a positive integer,

I

the

n \times n

identity matrix over

C

, and

J

the

2n \times 2n

matrix given by

J = \begin{bmatrix} 0 & I \\ -I & 0 \end{bmatrix}.

Let

M

be a

2n \times 2n

matrix over

C

of the form

M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}

where

A

,

B

,

C

,

D

are

n \times n

matrices over

C

. Find necessary and sufficient conditions on

A

,

B

,

C

,

D

in order that

M^t J M = J

.

Related theory

Group Frequencies And Functional Group Identification

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