# matrix psd inequalities

Let $$A,B,C$$ be real symmetric positive definite matrices. For some small $$\epsilon \ll 1$$, suppose $$(1-\epsilon)BB^{\top}\preceq AA^{\top} \preceq (1+\epsilon)BB^{\top}$$, do we have $$(1-\epsilon)BCB^{\top}\preceq ACA^{\top} \preceq (1+\epsilon)BCB^{\top}$$?

No. Let $$B=I$$ and $$A=\pmatrix{a&b\\ b&c}$$ be any positive definite matrix such that $$b\ne0$$ and $$(1-\epsilon)I\preceq A^2\preceq(1+\epsilon)I$$. Let $$C'=\operatorname{diag}(1,0)$$. Then $$AC'A=\pmatrix{a\\ b}\pmatrix{a&b}=\pmatrix{a^2&ab\\ ab&b^2}, \ (1+\epsilon)BC'B=\pmatrix{1+\epsilon&0\\ 0&0}.$$ Thus the inequality $$AC'A\preceq(1+\epsilon)BC'B$$ does not hold, because the bottom right entry of the LHS (i.e. $$b^2$$) is greater than the corresponding entry of the RHS (namely, $$0$$). Therefore it also doesn't hold when $$C$$ is any positive definite matrix that is sufficiently close to $$C'$$.