The Doubling Matrix and its Elementary Properties

Throughout we work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is in force in the dimensions $n$ and $2n=n+n$, for a natural number $n$ with $1\le n$. In particular $\iota:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{2n}$ is the concatenation map, a bijection, and the two-by-two array notation for block matrices is the one fixed in that clause; for $X,Y\in\mathcal{S}(n)$ we write $X\oplus Y\in\mathcal{S}(2n)$ for the block diagonal matrix they determine, and $-Y=(-1)Y$, an element of $\mathcal{S}(n)$ by Euclidean Space and Real Matrices: Standing Notation (iii). We write $2=1+1$, $3=2+1$ and $6=3+3$.

The doubling matrix in dimension $n$ is the block matrix

$$J=\begin{pmatrix} I_{n}&-I_{n}\\ -I_{n}&I_{n}\end{pmatrix},$$

where $-I_{n}=(-1)I_{n}$ as in Euclidean Space and Real Matrices: Standing Notation (iii).

Let $\xi,\eta\in\mathbb{R}^{n}$, let $X,Y\in\mathcal{S}(n)$, let $c\in\mathbb{R}$, and let $a\in\mathbb{R}$ be positive, so that $a^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. Then the following hold.

1. (Symmetry and action)(i) $J\in\mathcal{S}(2n)$ and

$$J\,\iota(\xi,\eta)=\iota(\xi-\eta,\ \eta-\xi);$$

in particular $J\,\iota(\xi,\xi)=0_{\mathbb{R}^{2n}}$.

2. (Quadratic form)(ii) $\iota(\xi,\eta)\cdot\bigl(J\,\iota(\xi,\eta)\bigr)=\lVert\xi-\eta\rVert^{2}$.

3. (Bounds)(iii) $0_{2n}\preceq J\preceq 2I_{2n}$ and $\lVert J\rVert\le 2$.

4. (Square)(iv) $J^{2}=2J$.

5. (Positive multiples)(v) $aJ\in\mathcal{S}(2n)$, $\lVert aJ\rVert\le 2a$, $3aJ\preceq 6aI_{2n}$, and

$$aJ+a^{-1}(aJ)^{2}=3aJ .$$

6. (Comparison of the diagonal blocks)(vi) If $X\oplus(-Y)\preceq cJ$, then $X\preceq Y$.