Matrix is invertible - proof. Tikhonov regularization and least squares problem.

Hello,

I have a big problem with proofing, that :

$\displaystyle A^TA+\lambda I$

is invertible.

$\displaystyle A^TA$ is invertible only if $\displaystyle rank(A^TA) = M$, where M is size of a matrix. Now i have to proof that, adding lambda times identity matrix will make the whole expression invertible.

I have a small advice from my professor: if A is a positive-definite matrix then there is a $\displaystyle A^(-1)$.

I've tried the standard approach $\displaystyle x \epsilon N(A^TA+\lambda I) \Rightarrow x^T(A^TA+\lambda I)x = x^T 0 $ but I've failed.

N means null space.

The whole task is to proof that least square with tikhonov regularization has only one exact solution, but I figure out that when above statement is invertible, we can compute exact solution.

I would really be great full, if someone could give me some advice.