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## Homework Statement

Let A be any nxn symmetric positive definite matrix. Show that (x‡0,xεRn)

x^TAx/x^Tx = the smallest eigenvalue of A.

## Homework Equations

## The Attempt at a Solution

Our hint was to first prove this for a diagonal matrix

For x^TAx/x^Tx I get L1x1² + L2x2² +...+Lnxn²/x1² + x2² +...+ xn² (I'm using L as lambda, the diagonal entries)

I know this is ≥1 since x1² + x2² +...+xn²/x1² + x2² +...+ xn² = 1 ≤ L1x1² + L2x2² +...+Lnxn²/x1² + x2² +...+ xn²

For the eigenvalues of A, If I choose x1, x2,.. to be 1 then xn = L1+L2+..L(n-1)/-Ln

I'm stuck here, help!