For $ A \in M_{m \times n}(\mathbb{R})$, does $\ker(A)$ relate to $\ker(A^T) $?

Question

For $ A \in M_{m \times n}(\mathbb{R})$, does $\ker(A)$ relate to $\ker(A^T) $? And if so, what would be the connection? I wouldn't imagine there being any without additional conditions on $A$ (like symmetry or something), but I'm not sure.

To provide some context, I'm trying to show that if $Ax = 0_m $ has a unique solution $x \in \mathbb{R^n}$, i.e. $\ker(A) = \{0_n\}$, then the matrix $A^TA$ is invertible. I reasoned that $Ax=0_m$ having a unique solution will not necessarily yield $\ker(A^TA) = \{0_n\}$ as $\ker(A^T)$ might not be $ \{0_m\}$. However, if $\ker(A) = \{0_n\}$ implies $\ker(A^T) = \{0_m\}$ then the solution should follow.

Thanks in advance,

Brandon

Answer 1

$\text{ker}(A)$ and $\text{ker}(A^T)$ are not related in general. However, $\text{ker}(A)$ and $\text{ker}(A^TA)$ do.

Lemma: $\text{ker}(A)=\text{ker}(A^TA)$.

Proof: It is obvious that $\text{ker}(A)\subset\text{ker}(A^TA)$.

Conversely let $x\in \text{ker}(A^TA)$, i.e. $A^TAx=0$. Premultiply with $x^T$ to get $$ x^T\underbrace{A^TAx}_{=0}=(Ax,Ax)=\|Ax\|^2=0\qquad\Rightarrow\quad Ax=0 \qquad\Rightarrow\quad x\in \text{ker}(A). $$ It gives $\text{ker}(A^TA)\subset\text{ker}(A)$, hence, equality.

Answer 2

The ranks of $A$ and $A^T$ are the same, and equal to the rank of $A^TA$ (this can be seen with singular value decomposition). Thus, if $m \leq n$, and $A$ is $m \times n$ and full rank $m$, then $A^TA$ will be invertible. However, if $m < n$ then $AA^T$ cannot be invertible, it is $n \times n$ and has rank $m$ where $m < n$.

Answer 3

Suppose $x \in \mathbf{R}^n$ and $A^T A x = 0$. Then \begin{align*} 0 &= \langle A^T Ax, x \rangle \\ &= \langle Ax, Ax \rangle, \end{align*} which implies that $Ax = 0$, which implies that $x = 0$ (because $Ker(A) = \{0\}$). Thus we have shown that $Ker(A^T A) = \{0\}$, which implies that the square matrix $A^TA$ is invertible.

Answer 4

You may be also interested in the following, familiar from the theory of Hilbert spaces operators, result: if $A:H \to H$ is some bounded operator on Hilbert space and $A^*$ denotes its adjoint then you have the decomposition $\ker(A) \oplus \overline{im(A^*)}=H$. If your space is finite dimensional, then $A$ can be viewed as a matrix (with respect to some orthogonal basis) and if all of the entries are real, then $A^*$ becomes the transposition $A^*=A^T$ so the above decomposition reads as $\ker(A) \oplus \overline{im(A^T)}=H$. So in this case $\ker(A)$ relates rather to the image of $A^T$

Answer 5

A very useful fact to be aware of is that if $A \in M_{m \times n}(\mathbb R)$ then $A$ and $A^TA$ have the same null space: \begin{align*} & Ax = 0 \\ \implies & A^T A x = 0 \\ \implies & x^T A^T A x = 0 \\ \implies & \| Ax \|^2 = 0 \\ \implies & Ax = 0. \end{align*}

Thus, if the null space of $A$ is trivial, then the null space of $A^T A$ is trivial, so $A^T A$ is invertible.