0
$\begingroup$

Let $A$ be an m×n matrix with $m$<$n$ and rank$(A) =m$. If $B=AA^T$ , $C=A^TA$ and the eigenvalues and corresponding eigenvectors of $B$ are known, find the non-zero eigenvalues and corresponding eigenvectors of $C$

My thinking: Since rank$(A)=m$, rank$(B)=m$, rank$(C) =m$. Hence $C$ has $m$ non zero eigenvalues. $A$ is not a square matrix. If an eigenvalue of $B$ is $\alpha$ and the corresponding eigenvector is $x$ then $Bx=\alpha x$. I can't proceed further. Please help me. Thank you in advance...

$\endgroup$
1
1
$\begingroup$

We have,

$$Bx = \alpha x \\ (AA^\top)x = \alpha x \\ A^\top (AA^\top) x = A^\top \alpha x \\ (A^\top A) A^\top x= \alpha A^\top x \\ CA^\top x = \alpha A^\top x $$ Let $A^\top x = q \ \in \mathbb{R}^{n}$. Then we have,

$$Cq = \alpha q$$ This shows that the eigenvectors of C are the eigenvectors of B multiplied by $A^\top$. What about the eigenvalues? We have $m$ distinct eigenvalues of B, but C has $n$ eigenvalues. Well as you guessed correctly, C is of rank(m), so it has $n-m$ zero eigenvalues, and the other $m$ eigenvalues are same as B.

$\endgroup$
2
  • $\begingroup$ What are the eigenvectors corresponding eigen value 0? $\endgroup$ May 6 at 17:51
  • $\begingroup$ To find the zero eigenvectors you are solving Cq = 0, so q is the null-space of C. $\endgroup$
    – orchi_d
    May 6 at 18:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.