Mean of L2 norm of random vector Ax+n

[cutesteph]

Homework Statement

What is the expected value of ||Ax+n|| where || || is the L2 norm and x and n are uncorrelated and E[n] = 0

The Attempt at a Solution

E[ norm of Y] = E[(Ax+n)' (Ax+n)] = E[(x'A'+n')(Ax+n)] = E[x'A'xA +x'T'n +n'Ax +n'n]
the three last terms = 0 due to uncorrelatedness so = E[x'A'xA] = E[tr(x'A'xA)] = E[tr(x'xA'A)] = A'AE[x'x] does this reduce any further?

 

[haruspex]

E[ norm of Y] = E[(Ax+n)' (Ax+n)] = E[(x'A'+n')(Ax+n)] = E[x'A'xA +x'T'n +n'Ax +n'n]

You mean E[x'A'Ax +x'A'n +n'Ax +n'n] , right?

 

[cutesteph]

Yes. But the result is still the same since it is a scalar and we can take the trace of it to rearrange. So does A'AE[x'x] simplify any further?

 

[haruspex]

Yes. But the result is still the same since it is a scalar

x'A'Ax is a scalar; x'A'x is a scalar; (x'A'x)A is an n x n matrix; x'(A'xA) is meaningless (since xA is meaningless).