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## Homework Statement

Show that the quadratic form [tex]h(x) = X^t(AA^t)X[/tex] is greater than or equal to 0 for all [tex]x[/tex] in [tex]R^n[/tex].

## Homework Equations

## The Attempt at a Solution

Since [tex](AA^t)^t = (A^t)^t A^t = AA^t[/tex], [tex]AA^t[/tex] can (according to the spectral theorem) be diagonalized in an orthonormal eigenvector basis. Assuming [tex]X = TX'[/tex] to be the relation between the bases, it follows that (since T is orthogonal):

[tex]h(x) = X^t(AA^t)X = (TX')^t(AA^t)(TX') = (X'^t T^t) (AA^t)(TX') = X'^t (T^{-1} AA^t T)X' = X'^t D X' = \mu_1 x_1'^2 + \mu_2 x_2'^2 + ... \mu_n x_n'^2[/tex]

The problem is thus equivalent to showing that [tex]\mu_1 \ge 0, \mu_2 \ge 0, ... , \mu_n \ge 0[/tex]. I haven't gotten anywhere with that approach.

I attempted to work directly on [tex]h(x) = X^t(AA^t)X[/tex] as well by trying to complete the square for 2x2, 3x3, ... matrices [tex]AA^t[/tex] (i.e. by using induction), and the proof would, if some relevant transformation is valid and all eigenvalues are ≥0, follow from Sylvester's law of inertia. I didn't get anywhere with that approach either.

Ideas?