Lets have a matix A of $n\times n$, in the follwing equation $A^3 + yA = O$ where y is a random number and O is a matrix of the same size with only zeros. May I replace a matrix with its eigenvalue in an equation and if so then why?
May I replace a matrix with its eigenvalue in an equation?
1
$\begingroup$
linear-algebra
eigenvalues-eigenvectors
2 Answers
3
$A^3+yA = O$
Let $\vec{v}$ be the eigenvector with $\lambda$ be the eigenvalue.
$$A^3\vec{v}+yAv=\vec{0}$$
$$(\lambda^3+y\lambda) \vec{v}=\vec{0}$$
Since $\vec{v} \neq \vec{0}$
$$\lambda^3+y\lambda = 0$$
0
Let $\mu$ be an eigenvalue of $A$, hence, with some $x \ne 0$:
$Ax= \mu x$. Then we get
$(A^3+yA)x= \mu^3x+y \mu x=(\mu^3+y \mu) x$. This shows that $\mu^3+y \mu$ is an eigenvalue of $A^3+yA$. Since $A^3+yA=0$, we get
$\mu^3+y \mu=0.$