- #1

pyroknife

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- 4

## Homework Statement

I am trying to prove the following:

if ##A\in C^{m\ \text{x}\ m}## is hermitian with positive definite eigenvalues, then A is positive definite. This was a fairly easy proof. The next part wants me to prove if A is positive definite, then ##\Delta_k##=\begin{bmatrix} a_{11} & \cdots & a_{1k} \\ \vdots & \ddots & \vdots\\ a_{k1} & \cdots & a_{kk} \end{bmatrix}

is also positive definite. k = 1...m

## Homework Equations

## The Attempt at a Solution

Since the first part I already proved if some matrix is hermitian with positive eigen values then that matrix is positive definite. I basically need to prove the ##\Delta_k## is hermitian with positive eigenvalues, therefore its positive definite.

It is obvious that ##\Delta_k## is hermitian since it is simply just a sub matrix of A, but how do I go about proving that this matrix also has positive eigenvalues? Any hints? Seems like it should be obvious, but I can't think of how to prove it.