Appending a row and column to a positive semi definite matrix

Question

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?

Answer 1

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

$$(x'b)^2-|b|^2 x'Ax \le 0 \iff |b|^2 x'Ax -x'bb'x \ge 0 \iff x'\left(A-\frac {bb'}{|b|^2}\right)x \ge0$$

Answer 2

Let $M$ be a Hermitian matrix over $\mathbb{C}$ partitioned into blocks as $$ M = \begin{pmatrix} A & B \\ B^* & C \end{pmatrix}, $$ where $A$ and $C$ are square matrices. Then the following are all equivalent

1. $M \geq 0$
2. $A \geq 0$, $(\mathbb{I} - A A^{-1})B = 0$ and $ C \geq B^* A^{-1} B$.
3. $ C \geq 0$, $(\mathbb{I} - C C^{-1})B^* = 0$ and $A \geq B C^{-1}B^*$.

where $A^{-1}$ and $C^{-1}$ are generalized inverses.

[1] - Zhang, Fuzhen (2005). The Schur Complement and Its Applications. Springer. doi:10.1007/b105056.