Proving that block diagonal matrix is orthogonal if its entries in the main diagonal are orthogonal matrix.

Question

Show that if $A,B$ are two orthogonal matrix then the block matrix $$P=\begin{pmatrix} A & 0 \\\ 0 & B \end{pmatrix}$$ is orthogonal.

Answer 1

Hint: Note that $$ {}^{\mathrm t\mkern-2mu}P=\begin{pmatrix} {}^{\mathrm t\mkern-2mu}A & 0 \\\ 0 & {}^{\mathrm t\mkern-2mu}B \end{pmatrix}$$ and use product by blocks.

Answer 2

Let

$A:V \to V$ and $B:W \to W$. The corresponding block diagonal matrix is exactly $(A \oplus B):V \oplus W \to V \oplus W$.

Hence, we have that the transpose is the matrix representation of the dual, so

$$\mathrm{Hom}(V \oplus W,k) =(V \oplus W)^* \cong(V^* \oplus W^*)$$ where the last isomorphism is the standard one used for $\mathrm{Hom}$, induced by restriction, so they have the same matrix representation. Hence,

$I_{V \oplus W}=I \oplus I=(AA^*) \oplus (BB)^*=(A \oplus B) \circ(A^* \oplus B^*) =(A \oplus B) \circ(A \oplus B)^*$