Symbolic matrices

Question

How does one declare a matrix of specific symbolic dimension? For instance, if $A,B,C,D$ are four $n\times n$ matrices, $M$ is the block matrix $$ M=\begin{bmatrix} A & B \\ C & D \end{bmatrix}, $$ and $J$ is the symplectic matrix $$ J=\begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}, $$ I would like to have Mathematica tell me that $$ MJM^\top-J=\begin{bmatrix} AB^\top-BA^\top & AD^\top-BC^\top -I_n \\ CB^\top - DA^\top + I_n & CD^\top - DC^\top \end{bmatrix}. $$

Answer 1

Define non-commutative multiplication of a pair of 2x2 block-matrices.

mul[{{a11_, a12_}, {a21_, a22_}}, {{b11_, b12_}, {b21_, b22_}}] :=
  {{a11 ** b11 + a12 ** b21, a11 ** b12 + a12 ** b22},
   {a21 ** b11 + a22 ** b21, a21 ** b12 + a22 ** b22}};

Define the transpose of a 2x2 block-matrix.

trans[{{a11_, a12_}, {a21_, a22_}}] := {{a11^T, a21^T}, {a12^T, a22^T}};

Define various simplification rules. There are more rules here than are needed to solve this problem.

Unprotect[NonCommutativeMultiply];
id ** u_ := u; u_ ** id := u;
u_ ** (-v_) := -u ** v; (-u_) ** v_ := -u ** v;
(u_ + v_) ** w_ := u ** w + v ** w; w_ ** (u_ + v_) := w ** u + w ** v;
0 ** u_ := 0; u_ ** 0 := 0;
Protect[NonCommutativeMultiply];

Define the block-matrices given in the original question.

m = {{a, b}, {c, d}};
j = {{0, id}, {-id, 0}};

Evaluate the expression given in the original question. This generates the result quoted in the original question.

mul[mul[m, j], trans[m]] - j // MatrixForm