signature of quadratic form and block matrix

Question

Suppose, there is a quadratic form with matrix $A = A^T$ and signature $(p,q)$. How can I find the signature of quadratic form with matrix $$ \begin{pmatrix} A & A \\ A & A \\ \end{pmatrix} $$ ?

I've tried to do block multiplication but I didn't get any useful results. Thanks for the help!

Answer 1

Suppose $B^tAB=D$, where $B$ is invertible and $D$ is diagonal, then $$\begin{pmatrix} B^t & -B^t \\ O & B^t \end{pmatrix}\begin{pmatrix} A & A \\ A & A \end{pmatrix} \begin{pmatrix} B & O \\ -B & B \end{pmatrix}=\begin{pmatrix} O & O \\ B^tA & B^tA \end{pmatrix}\begin{pmatrix} B & O \\ -B & B \end{pmatrix}=\begin{pmatrix} O & O \\ O & D \end{pmatrix}$$