0
$\begingroup$

I have a quick question I am trying to get the charecterstic polynimal of the following matrix using the principal minors of the matrix.

Using the following polynomial

$P_A(t)=t^4-E_1(A)t^3+E_2(A)t^2-E_3(A)t+E_4(A)$

I know the characteristic polynomial is

$P_A(t)=t^4-4t^3+3t^2+2t-1$

I know $E_1(A)=4$ is the trace

and $E_4(A)=-1$ the determinant

but I am having trouble choosing 4 principal minor in E_3(A) so they add up to -2.

$$ \begin{pmatrix} 1 & 1 & 0 &0 \\ 1 & 1& 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 &1 \end{pmatrix} $$

$\endgroup$
1
$\begingroup$

$E_k(A)$ is the sum of $k\times k$ principal minors of $A$, e.g., $E_3(A)=\det\left[\begin{matrix}1&1&0\\1&1&1\\0&1&1\end{matrix}\right]+0+0+\det\left[\begin{matrix}1&1&0\\1&1&1\\0&1&1\end{matrix}\right]=-1-1=-2$.

$\endgroup$

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.