# How to find eigenvalues and the corresponding eigenvectors

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...

$$Bx = \alpha x \\ (AA^\top)x = \alpha x \\ A^\top (AA^\top) x = A^\top \alpha x \\ (A^\top A) A^\top x= \alpha A^\top x \\ CA^\top x = \alpha A^\top x$$ 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.