Suppose I have a symmetric positive semi-definite matrix of size $(n \times n)$, $A$, and an $(n \times 1)$ dimensional column vector $b$. I construct the new matrix $\tilde{A}$ as follows:
$$\tilde{A}=\begin{bmatrix}A & b \\ b' & b'b\end{bmatrix}$$
Is there any way for me to tell if $\tilde{A}$ is now positive semidefinite? What conditions must $b$ satisfy in order for $\tilde{A}$ to be positive semi-definite?
Best Answer
Let consider $\tilde{x}=\begin{bmatrix}x \\t\end{bmatrix}$ then
$$\tilde{x}'\tilde{A}\tilde{x}=x'Ax+2x'bt+t^2b'b \ge 0$$
which requires
$$(x'b)^2-|b|^2 x'Ax \le 0 \iff |b|^2 x'Ax -x'bb'x \ge 0 \iff x'\left(A-\frac {bb'}{|b|^2}\right)x \ge0$$