Question

Let f(X) = −log|X| with the domain of f is the set of positive-definite matrices, denoted by S++. Notice that |X| is the determinant of matrix X. Show that f is convex.

Answer 1

since

f(x) = -log X

by the definition of convexivity we have to show that

f(xX + (1 - XY) < f(x) + (1 - 1)f(Y))
where $0< \lambda < 1$

Now,
log XX + (1 - XY > log(4X + (1 - XY)
{ because, det (A+B) $\geq$ Det(A) +Det (B) }

$=log|\lambda X|+log (|(1-\lambda )Y|/|\lambda X|)$ {USING , log mp = log m +log p}

$=(nlog\lambda +log| X|)+(nlog(1-\lambda)/\lambda ) +log (| Y|/| X|))$ WHERE n is the order of the matrix

  $\geq (\lambda log| X|)+(((1-\lambda )/\lambda )log (| Y|/| X|))$

  $\geq (\lambda log| X|)+(1-\lambda )log (| Y|/| X|)$ {because   }

(1/2-1) > (1 - )

> Vlog X + (1 - 1)log Y

therefore

-logAX + (1 - 1) <-Nog X - (1 - 1)log|Y|

i.e .

f(XX + (1 - XY)<Af(x) + (1 - 1)f(Y)

hence f is convex