Block Diagonal Symmetric Matrices and the Blocks of a Symmetric Matrix

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is fixed for every dimension and is used here with natural numbers $m$ and $n$ satisfying $1\le m$ and $1\le n$; we put $N=m+n$. The real numbers, the absolute value $|\cdot|$, the initial segments $[p]$, sequences, and Euclidean space with its sum and difference of points, dot product and Euclidean norm $\lVert\,\cdot\,\rVert$, the real matrices, their sums, differences and scalar multiples, the transpose, the matrix-vector product, the identity matrices $I_{p}$ and the zero matrices $0_{p\times q}$, the sets $\mathcal{S}(p)$ of symmetric real matrices, the positive semidefinite ordering $\preceq$, the norm $\lVert P\rVert$, the distance $d_{\mathcal{S}(p)}$ and the notions of convergence they determine and the concatenation map $\iota$ and the block matrix notation, are all as fixed there.

Let

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

be the concatenation map of Euclidean Space and Real Matrices: Standing Notation (vii), a bijection, for which every $k\in[N]$ satisfies exactly one of $k\in[m]$ and $k=m+j$ with $j\in[n]$ unique; the two-by-two array notation for block matrices is the one fixed in that clause. For $X\in\mathcal{S}(m)$ and $Y\in\mathcal{S}(n)$ write

$$X\oplus Y=\begin{pmatrix}X&0_{m\times n}\\0_{n\times m}&Y\end{pmatrix},$$

with the zero matrices $0_{m\times n}$ and $0_{n\times m}$ of Euclidean Space and Real Matrices: Standing Notation (ii).

Let $X,X'\in\mathcal{S}(m)$, let $Y,Y'\in\mathcal{S}(n)$, let $\xi\in\mathbb{R}^{m}$, let $\eta\in\mathbb{R}^{n}$ and let $a\in\mathbb{R}$. Then the following hold.

1. (Block diagonal matrices are symmetric) (i) $X\oplus Y\in\mathcal{S}(N)$; moreover $(X\oplus Y)-(X'\oplus Y')=(X-X')\oplus(Y-Y')$ and $(aI_{m})\oplus(aI_{n})=aI_{N}$.

2. (The quadratic form splits) (ii) $\iota(\xi,\eta)\cdot\bigl((X\oplus Y)\,\iota(\xi,\eta)\bigr)=\xi\cdot(X\xi)+\eta\cdot(Y\eta)$.

3. (Norm, distance and limits) (iii) $\lVert X\oplus Y\rVert$ is the larger of the two real numbers $\lVert X\rVert$ and $\lVert Y\rVert$, and consequently $d_{\mathcal{S}(N)}(X\oplus Y,X'\oplus Y')$ is the larger of $d_{\mathcal{S}(m)}(X,X')$ and $d_{\mathcal{S}(n)}(Y,Y')$. Hence, for sequences $(X_{k})_{k\in\mathbb{N}}$ in $\mathcal{S}(m)$ and $(Y_{k})_{k\in\mathbb{N}}$ in $\mathcal{S}(n)$, the sequence $(X_{k}\oplus Y_{k})_{k\in\mathbb{N}}$ converges to $X\oplus Y$ in $\mathcal{S}(N)$ if and only if $(X_{k})$ converges to $X$ in $\mathcal{S}(m)$ and $(Y_{k})$ converges to $Y$ in $\mathcal{S}(n)$.

4. (The ordering splits) (iv) $X\oplus Y\preceq X'\oplus Y'$ if and only if both $X\preceq X'$ and $Y\preceq Y'$.

5. (The blocks of a symmetric matrix) (v) Let $Z\in\mathcal{S}(N)$ and define real matrices $Z^{11}$, $Z^{12}$ and $Z^{22}$, of sizes $m\times m$, $m\times n$ and $n\times n$, by

$$(Z^{11})_{ij}=Z_{ij},\qquad (Z^{12})_{ij}=Z_{i,\,m+j},\qquad (Z^{22})_{ij}=Z_{m+i,\,m+j},$$

the indices ranging over $i,j\in[m]$ in the first case, over $i\in[m]$ and $j\in[n]$ in the second, and over $i,j\in[n]$ in the third. Then $Z^{11}\in\mathcal{S}(m)$, $Z^{22}\in\mathcal{S}(n)$,

$$Z=\begin{pmatrix}Z^{11}&Z^{12}\\ (Z^{12})^{\top}&Z^{22}\end{pmatrix},$$

and $Z=Z^{11}\oplus Z^{22}$ if and only if every entry of $Z^{12}$ equals $0$. Moreover

$$\iota(\xi,\eta)\cdot\bigl(Z\,\iota(\xi,\eta)\bigr)=\xi\cdot(Z^{11}\xi)+2\,\xi\cdot(Z^{12}\eta)+\eta\cdot(Z^{22}\eta),$$

where $2$ denotes $1+1$.