Let $n\in \mathbb{N}$, and $A\in \mathbb{R}^{n\times n}$ be a semi-positive definite matrix. What can we say about the matrix
$$ A_h:= A+(h-1)I, $$
where $I$ is the identity matrix, and $h>0$ (interested here in small $h$)?
I believe there will be an $h_0$ such that for all $h<h_0$, $A_h$ is invertible. But I wonder if there is anything else we can say about $A_h$, can I get lower (and upper) bounds on $\langle A_h(x-y),x-y\rangle$ ?
We can say a lot about $A_h$ since the identity matrix $I$ commutes with every other matrix. For simplicity I assume that $A$ is diagonizable.
Thus there is an orthogonal matrix $B$ such that $BAB^{-1}=D$, where $D$ is a diagonal matrix with the eigenvalues of $A$ on its diagonal. Then we have $$ BA_hB^{-1}=BAB^{-1}+(h-1)BIB^{-1}=D+(h-1)I\, ,$$ since $BI=IB$.
In conclusion: we used that $A_h$ has the same eigenbasis as $A$ and saw that the eigenvalues of $A_h$ are given by the entries on the diagonal of $D_h:=D+(h-1)I$. $A_h$ is invertible if $D_h$ is and the eigenvalues will give you a lower and upper bound on $\langle A_hv,v\rangle$ aswell.