Does the Matrix xyTA Have More Than One Non-Zero Eigenvalue?

[Leo321]

We have vectors x,y of size n and a matrix A of size nxn.
Is it true that the matrix xy^TA has at most one non zero eigenvalue? Why is it so?

 

[Leo321]

Ok, I think I got it.
The matrix xy^T has a rank of 1.
It is known that rank(AB)<=min(rank(A),rank(B))
Thus rank(xy^TA)<=1
It is also known that the number of non-zero eigenvalues of a matrix is less or equal to the matrix rank. Thus the number of non-zero eigenvalues of xy^TA is at most 1.
Right?

 

[khemist]

xy(transpose) will yield a scalar correct?

That means the maximum number of eigenvalues is n i believe

 

[Outlined]

xy(transpose) will yield a scalar correct?

no, x^Ty is a scalar