Eigenvalues of matrix

[maria23]

need help with this question:
let A=(a₁, a₂,...,a_n) be row vector, find the eigenvalues of B=A^tA...
i calculated the characteristic polynomial of B... I got nothing
i got A^tA=
a₁a₁ a₁a₂..................a_na₁
a₁a₂ a₂a₂.................
...
...
a₁a_n ....................... a_na_n

i dont know what to do with that

 

[HallsofIvy]

If you are actually trying to find a general solution that is an extremely hard question! It might help to look at the n= 2 and n= 3 cases.

If n= 2, the matrix is \(\displaystyle \begin{bmatrix}a_1^2 & a_1a_2 \\ a_1a_2 & a_2^2\end{bmatrix}\).

The eigenvalue equation is \(\displaystyle \left|\begin{array}{cc}a_1^2- \lambda & a_1a_2 \\ a_1a_2 & a_2^2-\lambda\end{array}\right|\)\(\displaystyle = (\lambda- a_1^2)(\lambda- a_2^2)- a_1^2a_2^2\)\(\displaystyle = \lambda^2- (a_1^2+ a2^2)\lambda= 0\).

What are the roots of that?

 

[maria23]

..

thnx... thats the thing, i want it general... i can do it for specific matrix
I can give u the general eigenvalues equation.. I dont know what to do with it