Elementwise maximum of two positive definite matrices

Question

Assume that $A$ and $B$ are real, symmetric, positive definite matrices of the same size, that is,

$$A \succ 0, B\succ 0.$$

Let $\operatorname{elmax(A,B)}$ be the element-wise maximum matrix, consisting of scalar maxima of elements of $A$ and $B$.

Is it always the case that

$$\operatorname{elmax}(A,B)\succ 0?$$

That is, does the element-wise maximum operation preserve positive definiteness?

Answer 1

No. A conterexample: take a small $\epsilon>0$ and define $$ A=\begin{bmatrix}1 & 1-\epsilon & 0\\1-\epsilon & 1 & 0\\0 & 0 & 1\end{bmatrix},\qquad B=\begin{bmatrix}1 & 0 & 0\\0 & 1 & 1-\epsilon\\0 & 1-\epsilon & 1\end{bmatrix}. $$ Both matrices are positive definite, but the element wise maximum $$ C=\begin{bmatrix}1 & 1-\epsilon & 0\\1-\epsilon & 1 & 1-\epsilon\\0 & 1-\epsilon & 1\end{bmatrix} $$ has a negative determinant (for small $\epsilon>0$).

Answer 2

I typed almost the same answer A.Γ. so here is a concrete example instead

A
array([[22.,  9.,  7.,  6.],
       [ 9., 14.,  5.,  8.],
       [ 7.,  5., 12., 11.],
       [ 6.,  8., 11., 14.]])

B
array([[20.,  2., 12.,  2.],
       [ 2., 20.,  2.,  5.],
       [12.,  2., 12.,  4.],
       [ 2.,  5.,  4., 12.]])

C
array([[22.,  9., 12.,  6.],
       [ 9., 20.,  5.,  8.],
       [12.,  5., 12., 11.],
       [ 6.,  8., 11., 14.]])

with eigenvalues

eigA = [ 1.30323419,  7.88111427, 13.88897809, 38.92667345]
eigB = [ 2.81884072,  9.73820335, 20.42843182, 31.01452411]
eigC = [-0.11100762, 11.43648115, 13.68814039, 42.98638609]