Verify Proposition B.2 in the special case of 1+5, 2+4,3+3 blocks in a 6x6 matrix.

Block 1+5

Q=(({{0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}}));

P=(({{a1, 0, 0, 0, 0, 0}, {0, b11, b12, b13, b14, b15}, {0, b21, b22, b23, b24, b25}, {0, b31, b32, b33, b34, b35}, {0, b41, b42, b43, b44, b45}, {0, b51, b52, b53, b54, b55}}));

Inverse[Q].P.Q//MatrixForm

Union[Flatten[Transpose[Q]-Inverse[Q]]]//MatrixForm

Block 2+4

Q=(({{0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}}));

P=(({{a1, a2, 0, 0, 0, 0}, {a3, a4, 0, 0, 0, 0}, {0, 0, b1, b2, b3, b4}, {0, 0, b5, b6, b7, b8}, {0, 0, b9, b10, b11, b12}, {0, 0, b13, b14, b15, b16}}));

Inverse[Q].P.Q//MatrixForm

Union[Flatten[Transpose[Q]-Inverse[Q]]]//MatrixForm

Block 3+3

Q=(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}}));

P=(({{a1, a2, a3, 0, 0, 0}, {a4, a5, a6, 0, 0, 0}, {a7, a8, a9, 0, 0, 0}, {0, 0, 0, b1, b2, b3}, {0, 0, 0, b4, b5, b6}, {0, 0, 0, b7, b8, b9}}));

Inverse[Q].P.Q//MatrixForm

Union[Flatten[Transpose[Q]-Inverse[Q]]]//MatrixForm