Two questions on the Moore–Penrose inverse

Question

We define the Moore–Penrose inverse of a matrix $A$ as follows

$$A^+ := \lim_{x \to 0} \, (A^T A + xI)^{-1} A^T$$

and we say that it finds the lowest norm solution of $\|Ax - y\|$. I'm not sure about 2 things:

1. Why is $(A^T A + xI)$ always invertible?

2. How does this definition imply $A^+ A = I$?

Answer 1

Sketch for proof of 1): if $\lambda_1,\ldots,\lambda_n$ are the eigenvalues of $A^\top A$, then $\lambda_1+x, \ldots, \lambda_n + x$ are the eigenvalues of $A^\top A + x I$ (why?). Since the $\lambda_i$ are nonnegative (why?) and $x > 0$, we see that $A^\top A + x I$ has no zero eigenvalues.

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The pseudoinverse does not satisfy $A^+ A = I$. Check the definition/properties again.