When do some elements of the null space of a matrix always be equal to 0?

I have an underdetermined system of the form

$\begin{bmatrix} I& M_{12} \\ M_{21} & M_{22} \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \ 0$

with more rows than columns, and the $1 \times 1$ block $I$ is the identity matrix of dimensions $n \times n$, $M_{12}$ is a block diagonal matrix of dimension $n \times m$, and $M_{21} \in R^{p \times n}$ and finally $n+p<n+m$. Moreover, $n<m$, and $x_1 \in R^n$ and $x_2 \in R^m$.

I am trying to identify the conditions that matrices $M_{12},M_{21},M_{22}$ must satisfy so that in the null space of this system, $x_1=0$ is the only possibility.

I have been able to work out examples using the row-echelon form, but a more general result would be very much useful.

Thank you very much!

Solution 1:

We can apply a block-row operation to the matrix as follows: $$ \pmatrix{I & 0 \\-M_{21} & I} \pmatrix{I & M_{12}\\ M_{21} & M_{22}} = \pmatrix{I & M_{12}\\ 0 & M_{22} - M_{21}M_{12}}. $$ In other words, the system of equations can be rewritten as $$ \begin{cases} (M_{22} - M_{21}M_{12})x_2 = 0,\\ x_1 = -M_{12} x_2. \\ \end{cases} $$ So, the solutions to this equation will all satisfy $x_1 = 0$ if and only if $(M_{22} - M_{21}M_{12})x_2 = 0$ implies that $M_{12}x_2 = 0$. Expressed more concisely, we require that the nullspace of $M_{12}$ contains the nullspace of $(M_{22} - M_{21} M_{12})$.

With that, there are a few approaches one may take to check this condition. An approach that I suspect you're more likely to be comfortable with is to begin by finding a basis $\{v_1,\dots,v_k\}$ for the nullspace of $M_{22} - M_{21}M_{12}$. The desired condition holds if and only if we have $M_{12}v_j = 0$ for $j = 1,\dots,k$. Another approach is to use the Moore-Penrose pseudoinverse. The desired condition holds if and only if $$ M_{12}[I - A^+A] = 0, $$ where $A = M_{22} - M_{21}M_{12}$. If $A$ has linearly independent rows, then we may write $A^+A = A^T(AA^T)^{-1}A$.