Loewner order

Remarks

The relation $\succeq$ is a partial order. It is transitive because sums of positive semidefinite matrices are positive semidefinite. It is antisymmetric because a matrix that is both positive semidefinite and negative semidefinite is $0$. Example: $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \succeq I$, because the difference $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ has eigenvalues $2$ and $0$. Non-example: $\operatorname{diag}(1, 0) \not\succeq \operatorname{diag}(0, 1)$, because the difference $\operatorname{diag}(1, -1)$ has the eigenvalue $-1$; the converse fails too, so the order is not total. The order does not respect squares: for $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $A \succeq B$, but $A^2 - B^2 = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}$ has determinant $-1$. On the other hand, $A \succeq B \succeq 0$ implies $A^{1/2} \succeq B^{1/2}$.