Tessellate S.T.E.M.S - Mathematics - College - Set 3 - Problem 1

Algebra Level 5

Let $S \in M_{2n} (\mathbb{R}), n \in \mathbb{Z}^{+}$ such that S contains exactly the elements $A \in M_{2n}(\mathbb{R})$ which satisfy $A I_n' = I_n' A$ where $\displaystyle I_n' = \left( \begin{array}{c|c} 0_{n,n} & -I_n \\ \hline -I_n & 0_{n,n}\\ \end{array} \right)$

Find the dimension of $S$ when considered as a vector space over $\mathbb{R}$ .

Details and assumptions :

$I_n$ and $0_{n,n}$ denote the identity matrix and zero matrix of size $n\times n$ , respectively.
$M_{2n}(\mathbb{R})$ is the set of all matrices of size $2n\times 2n$ with real entries.

This problem is a part of Tessellate S.T.E.M.S.

4n

2n

2n^2

n^2

1 solution

Patrick Corn
Jan 17, 2018

A block matrix $M = \left( \begin{array}{(c|c)} A & B \\ \hline C & D \\ \end{array} \right),$ where $A,B,C,D$ are $n\times n$ matrices, satisfies $\begin{aligned} M I_n' &= \left( \begin{array}{(c|c)} -B & -A \\ \hline -D & -C \\ \end{array} \right) \\ I_n' M &= \left( \begin{array}{(c|c)} -C & -D \\ \hline -A & -B \\ \end{array} \right) \end{aligned}$ These are equal if and only if $A=D$ and $B=C.$ So we can choose whatever $A,B$ we want, and $C$ and $D$ are uniquely determined. The dimension of the corresponding vector space is $n^2+n^2 = 2n^2.$

Tessellate S.T.E.M.S - Mathematics - College - Set 3 - Problem 1

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