If A and B are real matrices, with A being symmetric, B having at least as many columns as rows, and
the matrix C defined as:
$$
\begin{bmatrix}
A & B^T \\
B & 0 \\
\end{bmatrix}
$$
how can I prove that:
1) C is invertible, if A is positive definite and B of full rank
and,
2) Is C always invertible, if A is invertible and B of full rank?
My attempt so far was to sketch the block matrices.
For part 1)
I let A be of size $nxn$. Since A is positive definite, it is invertible, and thus has full rank.
I let B be of size $(k-(n+1)+1)x(p)$ = $(k-n)x(p)$, so that $B^T$ is of size $(p)x(k-n)$. Then C is of size $kxk$. Then I try to argue that, this $kxk$ square matrix has full rank, with rank = k, which implies that C is invertible. B has full row rank, and $B^T$ has full column rank.
Am I sort of close to the answer? I'm basically trying to avoid the usage of determinants of block matrices, as I'm not all that comfortable with that method – but perhaps it's necessary for this question.
For part 2)
My work for part 1), if it's correct, would imply that, yes, C is always invertible if A is invertible and B of full rank. I get the feeling, though, that there is a counterexample.
Thanks in advance for your help,
Best Answer
Since $A$ is nonsingular, consider the following block factorization of $C$: $$ C=\pmatrix{A&B^T\\B&0} = \pmatrix{I&0\\BA^{-1}&I}\pmatrix{A&0\\0&S}\pmatrix{I&A^{-1}B^T\\0&I}, $$ where $S:=-BA^{-1}B^T$. Since the triangular blocks are nonsingular, the matrix $C$ is nonsingular iff the Schur complement matrix $S$ is nonsingular.
Now if $A$ is SPD, it is easy to see that $S$ is SPD as well. First, the definiteness of $A$ implies that $A^{-1}$ is SPD. For a nonzero $x$, $B^Tx\neq 0$ since has $B$ has full row rank, and $$x^T(BA^{-1}B^T)x=(B^Tx)^TA^{-1}(B^Tx)>0.$$
Another way to see that $C$ is nonsingular if $A$ is SPD and $B$ has full row rank is as follows. Assume that $Cz=0$ for some nonzero $z=(x^T,y^T)^T$. Hence $$\tag{1} Ax+B^Ty=0, \quad Bx=0. $$
None of the block components can be zero. If $x=0$ and $y\neq 0$ then $B^Ty=0$ which is impossible since $B$ has full row rank. If $x\neq 0$ and $y=0$ then $Ax=0$ which is impossible since $A$ is SPD. Hence both $x\neq 0$ and $y\neq 0$. Multiply the first equation in (1) with $x^T$ and the second with $y^T$ to get $$ x^TAx+x^TB^Ty=0, \quad y^TBx=0. $$ Since $x^TB^Ty=y^TBx=0$, we have $x^TAx=0$, which gives again a contradiction.
It is not sufficient that $A$ is nonsingular and $B$ of full rank for $S$ being nonsingular. Consider, $$ A=\pmatrix{1 & 0 \\ 0 & -1}, \quad B=(1,1). $$ It is easy to verify that $C$ is singular (actually $S=0$).