Eigenvalue

In linear algebra an eigenvalue of a (square) matrix $A$ is a number $\lambda$ that satisfies the eigenvalue equation,

${\text{det}}(A-\lambda I)=0\ ,$

where $I$ is the identity matrix of the same dimension as $A$ and in general $\lambda$ can be complex. The origin of this equation is the eigenvalue problem, which is to find the eigenvalues and associated eigenvectors of $A$. That is, to find a number $\lambda$ and a vector ${\vec {v}}$ that together satisfy

$A{\vec {v}}=\lambda {\vec {v}}\ .$

What this equation says is that even though $A$ is a matrix its action on ${\vec {v}}$ is the same as multiplying it by the number $\lambda$. Note that generally this will not be true. This is most easily seen with a quick example. Suppose

$A={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}$ and ${\vec {v}}={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}\ .$

Then their matrix product is

$A{\vec {v}}={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}={\begin{pmatrix}a_{11}v_{1}+a_{12}v_{2}\\a_{21}v_{1}+a_{22}v_{2}\end{pmatrix}}$

whereas

$\lambda {\vec {v}}={\begin{pmatrix}\lambda v_{1}\\\lambda v_{2}\end{pmatrix}}\ .$