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Convexity of exponential function

If are symmetric real matrices we write if-f the matrix is non negative definite. Examine if

for each pair real symmetric matrices such that .

Solution

Since commute it follows from the properties of the exponential function

(1)

Noting that

(2)

Setting we must prove that . Since commute so are . Hence is symmetric. Thus,

(3)

and that’s all!

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