Action and Quadratic Form of a Block Matrix

Let $m$ and $n$ be natural numbers with $1\le m$ and $1\le n$, let $\mathbb{R}$ be the real numbers, and for a natural number $p$ let $[p]$ be the initial segment determined by $p$. Let $A$, $B$, $C$ and $D$ be real matrices of sizes $m\times m$, $m\times n$, $n\times m$ and $n\times n$, and let $M$ be the block matrix they determine.

Regard Euclidean space $\mathbb{R}^p$ as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the sum $z+z'$ of points and the dot product $z\cdot z'$; write $Nz$ for the matrix-vector product, $N^{\top}$ for the transpose, and let

$$\iota:\mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}^{m+n}$$

be the concatenation map of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space. Let $\xi,\xi'\in\mathbb{R}^m$ and $\eta,\eta'\in\mathbb{R}^n$.

Then the following hold.

1. (Action)

$$M\,\iota(\xi,\eta)=\iota\bigl(A\xi+B\eta,\ C\xi+D\eta\bigr).$$

2. (Bilinear form)

$$\iota(\xi',\eta')\cdot\bigl(M\,\iota(\xi,\eta)\bigr)=\xi'\cdot(A\xi)+\xi'\cdot(B\eta)+\eta'\cdot(C\xi)+\eta'\cdot(D\eta).$$

3. (Symmetry) $M$ is symmetric if and only if $A=A^{\top}$, $D=D^{\top}$ and $C=B^{\top}$.