TT* + I is invertible

I've the following exercise which I can't solve:

Prove that:

$$AA^* + I$$

is invertible for all Matrix $A$ in finite-dimensional field $V$ with inner product. $A^*$ is the adjoint operator.

Any help will be appreciated.

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But it is.${{}}$ – Git Gud Jul 7 '14 at 13:21
Oops, my mistake. It should is invertible, I fixed the title. – danny11 Jul 7 '14 at 13:25
How is the adjoint defined for finite fields, where there isn't really an inner product? – Nishant Jul 7 '14 at 13:26
V is inner product space. – danny11 Jul 7 '14 at 13:29
Simple question always gets many answers LoL – Mr.T Jul 7 '14 at 13:46

$<AA^*v+v,v>=<AA^*v,v>+<v,v>=<A^*v,A^*v>+<v,v>$

this is true from the property of inner product and definition of $A^*$.

Let's call $A^*v$ in another name, let's call it the vector $\alpha$.

So

$<AA^*v+v,v>=<A^*v,A^*v>+<v,v>=<\alpha,\alpha>+<v,v>=\|\alpha\|+\|v\|$

if $v \neq 0$, then this term is larger than zero. Which means $AA^*+I$ is positive definite, which means it only has positive eigenvalues, which means 0 isn't an eigenvalue, which means it is invertible.

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Assume that there's a $v\in V$ such that $AA^*v=-v$. Can you use the definition of the adjoint to conclude that $v=0$?

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If $A^*A + I$ is not invertible then $det(A^*A + I)=0$ then $-1$ is an eigenvalue of $A^*A$ then there exist nonzero $v$ such that $A^*Av=-v$ therefore $$v^*A^*Av=-v^*v$$ which means nonsense because norm is not negative.

Then $A^*A + I$ is always invertible.

Why does $v$ always imply column-vector and $v^*$ always imply row-vector?

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Assuming that there is an inner product in $V$. Then for every $x \in V, x \neq 0$, we have $$x^*(AA^*+I)x = \underbrace{(A^*x)^*(A^*x)}_{\geq 0}+\underbrace{x^*x}_{>0}$$ So $AA^*+I$ is positive definite and thus invertible.

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