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Operator Monotonicity and Convexity

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Operator Monotonicity and Convexity

Vjosa Blakaj Homework of Topic 5

May 19, 2018

Problem 1.

1. Show that the square root function is matrix monotone in a different way from that presented in class.

2. Lowener Theorem tells us that the function f(x) = xt for every 0 ≤ t ≤ 1 is matrix monotone. Does the monotonicity still holds for t >1?

Problem 2.

Show that the functionf(x) := exp(x) is not matrix monotone.

Hint: Use the divided difference matrix.

Problem 3.

Show that the map (A, B)7→M0(A, B) on Hn+ x Hn+ is jointly concave.

Hint: Use the identity

M0(A, B) = max{C |

A C

C B

≥0} (1)

1

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