Properties of $\mathbf{A}^T\mathbf{A}$ and $\mathbf{A}\mathbf{A}^T$?

Question

What are the differences with regards to properties of $\mathbf{A}^T\mathbf{A}$ and $\mathbf{A}\mathbf{A}^T$? For example, the former has those mentioned here. Is the latter not positive (semi)definite as well? Will there always be eigenvalues for both of the matrix products?

In terms of the properties you mention, you should think of $B=A^T$, then $AA^T=B^T B$ and so they have the same properties. They also have joint properties (same eigen values, trace and determinant among others). — P. Quinton, Sep 24 '20 at 19:45
@P.Quinton but isn't this a counter example: Let $\mathbf{A} = \left[\begin{array}{cc} 2&1 \end{array}\right]$. $\mathbf{A}\mathbf{A}^T$ does not have any eigenvalues(it is a scalar) but $\mathbf{A}^t\mathbf{A}$ is a matrix with eigenvalues 5 and 0? — That Guy, Sep 24 '20 at 20:06
@P.Quinton Can you or anyone please help me with the question in the comment above?` — That Guy, Sep 25 '20 at 00:34

score 0 · Accepted Answer · answered Sep 24 '20 at 20:21

Here are some properties:

Property 1. $AA^T$ and $A^TA$ are both symmetric.

Property 2. $\mathrm{rank}(A)=\mathrm{rank}(AA^T)=\mathrm{rank}(A^TA)$

If $Ax=0$ then $A^TAx=0$. If $A^TAx=0$, then $x^TA^TAx=0$, i.e. $||Ax||^2=0\Rightarrow Ax=0$. Thus $\mathrm{nullity}(A)=\mathrm{nullity}(A^TA)$. Interchanging $A$ and $A^T$, we have $\mathrm{nullity}(A)=\mathrm{nullity}(A^T)=\mathrm{nullity}(AA^T)$.

Property 3. $AA^T$ and $A^TA$ have the same non-zero eigenvalues.

If $\lambda$ is an eigenvalue of $A^TA$, then for some $v$, $A^TAv=\lambda v\Rightarrow AA^T(Av)=\lambda(Av)$. For the converse exchange $A^T$ and $A$.

We also see that $$\mathcal{C}(A^TA)\stackrel{(1)}{=}\mathcal{R}(A^TA)\stackrel{(2)}{=}\mathcal{R}(A)\ \text{and}\ \mathcal{R}(AA^T)\stackrel{(1')}{=}\mathcal{C}(AA^T)\stackrel{(2')}{=}\mathcal{C}(A),$$ where $(1)$ and $(1')$ follows from Property 1 (For a symmetric matrix $M$, $\mathcal{C}(M)=\mathcal{R}(M^T)=\mathcal{R}(M)$)

and $(2)$ and $(2')$ follows from the Property 2 (For any two matrix $M$ and $N$, $\mathcal{R}(MN)\subseteq\mathcal{R}(N)$ and $\mathcal{C}(MN)\subseteq\mathcal{C}(M)$. If $W$ is a subspace of $V$ and $\mathrm{dim}(W)=\mathrm{dim}(V)$, then $WW=V$).

Properties of $\mathbf{A}^T\mathbf{A}$ and $\mathbf{A}\mathbf{A}^T$?

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