Let $A$ be an $n\times n$ real symmetric matrix, and $\operatorname{sign}(A)$ is the matrix sign function, which is defined as $\operatorname{sign}(A) := Z \pmatrix{-I_p & 0\cr 0 & I_q\cr}Z^{-1}$, if $A = ZJZ^{−1}$ is a Jordan canonical form arranged so that $J = \operatorname{diag}(J_1, J_2)$, where the eigenvalues of $J_1 ∈ \Bbb R^{p\times p}$ lie in the open left half-plane and those of $J_2 ∈ \Bbb R^{q\times q}$ lie in the open right half-plane.
The matrix sign function may also be calculated as $\operatorname{sign}(A) = A(A^2)^{-1/2}$.
(Obviously, the above definition holds for the matrix $A$, if $A$ has now eigenvalue on the imaginary axis.)
Is true to conclude (similar to the scalar case) that $A \leq \operatorname{sign} (A) \times A = A\times \operatorname{sign}(A) $ ?
In addition, by $A\leq B$ we mean that $A-B$ is a positive semi-definite matrix.