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CyberLink PowerDirector Ultimate Suite 17.0.2420.0 + Crack Download PC. CyberLink PowerDirector Ultimate Suite 17.0.2420.0 + Crack free .Q:
Question on eigenvalues and eigenvectors of a matrix
Suppose we have a matrix $A \in \Bbb R^{nxn}$ such that
$$
A = \begin{bmatrix}
-1 & 3 & 3 \\
1 & -3 & -3 \\
3 & 1 & -1 \\
\end{bmatrix}
$$
I want to find all the eigenvalues and the corresponding eigenvectors.
I understand that the characteristic polynomial of the matrix is
$$\chi_A(\lambda) = (\lambda+1)^3-3\lambda^2-3\lambda$$
And I found out that the eigenvalues are $-1\pm i, -1-i$
My question is, how do you find the eigenvectors corresponding to these two eigenvalues?
A:
If $x$ is an eigenvector then so are $c_1x, c_2x$ with $c_1,c_2 \in \mathbb R$. Since $A(c_1x+c_2x) = c_1A(x)+c_2A(x) = c_1(-1-i)+c_2(1+i) = -c_1+c_2$ then
$$
c_1A(x) = -c_1+c_2 \\
c_2A(x) = c_1-c_2
$$
and, from the second of these, we see that $c_2 = \overline{c_1}$.
So $\{x,-\overline{x}\}$ is a basis for the eigenspace corresponding to the eigenvalue $-1-i$ and $\{x,\overline{x}\}$ is a basis for the eigenspace corresponding to the eigenvalue $1+i$.
You’ll probably
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