Testing a Block Semidefinite Inequality on the Diagonal

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers, let $\alpha\in\mathbb{R}$, and let $X$ and $Y$ be symmetric real $n\times n$ matrices. Write $I_n$ for the identity matrix of size $n$, write $0_n$ for the real $n\times n$ matrix all of whose entries are the additive identity $0$ of $\mathbb{R}$, and write $\mu P$ for the scalar multiple of a matrix $P$; set $-Y=(-1)Y$ and $-\alpha I_n=(-\alpha)I_n$. The matrices $0_n$ and $I_n$ are symmetric, their entries being unchanged when the two indices are interchanged, and hence so are $-Y$, $\alpha I_n$ and $-\alpha I_n$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure; moreover $0_n^{\top}=0_n$ and $(-\alpha I_n)^{\top}=-\alpha I_n$ with the transpose.

Let

$$M=\begin{pmatrix}X&0_n\\0_n&-Y\end{pmatrix},\qquad N=\begin{pmatrix}\alpha I_n&-\alpha I_n\\-\alpha I_n&\alpha I_n\end{pmatrix}$$

be the block matrices determined by these blocks; both are symmetric real $(n+n)\times(n+n)$ matrices by claim 3 of Action and Quadratic Form of a Block Matrix. Write $P\preceq Q$ for the positive semidefinite ordering.

If $M\preceq N$, then $X\preceq Y$.