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 det means the determinant, $I$ is the identity matrix of the same dimension as $A$, and in general $\lambda$ can be complex. The origin of this equation, the characteristic polynomial of A, 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 $\scriptstyle {\vec {v}}$ that together satisfy

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

What this equation says is that even though $A$ is a matrix its action on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle\vec{v}} is the same as multiplying the vector by the number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} . This means that the vector $\scriptstyle {\vec {v}}$ and the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle A\vec{v}} are parallel (or anti-parallel if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is negative). 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 the scalar product is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda\vec{v}=\begin{pmatrix} \lambda v_1 \\ \lambda v_2 \end{pmatrix}\ .}

Obviously then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle A\vec{v}\neq \lambda\vec{v}} unless Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda v_1 = a_{11}v_1+a_{12}v_2} and simultaneously $\lambda v_{2}=a_{21}v_{1}+a_{22}v_{2}$. For a given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} , it is easy to pick numbers for the entries of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and $\scriptstyle {\vec {v}}$ such that this is not satisfied.

The eigenvalue equation

So where did the eigenvalue equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det}(A-\lambda I)=0} come from? Well, we assume that we know the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and want to find a number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} and a non-zero vector $\scriptstyle {\vec {v}}$ so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle A\vec{v}=\lambda\vec{v}} . (Note that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle\vec{v}=\vec{0}} then the equation is always true, and therefore uninteresting.) So now we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle A\vec{v}-\lambda\vec{v}=\vec{0}} . It doesn't make sense to subtract a number from a matrix, but we can factor out the vector if we first multiply the right-hand term by the identity, giving us

$(A-\lambda I){\vec {v}}={\vec {0}}\ .$

Now we have to remember the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A-\lambda I} is a square matrix, and so it might be invertible. If it was invertible then we could simply multiply on the left by its inverse to get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=(A-\lambda I)^{-1}\vec{0}=\vec{0}}

but we have already said that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle\vec{v}} can't be the zero vector! The only way around this is if $A-\lambda I$ is in fact non-invertible. It can be shown that a square matrix is non-invertible if and only if its determinant is zero. That is, we require

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det}(A-\lambda I)=0\ ,}

which is the eigenvalue equation stated above.

A more technical approach

So far we have looked eigenvalues in terms of square matrices. As usual in mathematics though we like things to be as general as possible, since then anything we prove will be true in as many different applications as possible. So instead we can define eigenvalues in the following way.

Definition: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a vector space over a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , and let $\scriptstyle A:V\to V$ be a linear map. An eigenvalue associated with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is an element $\scriptstyle \lambda \in F$ for which there exists a non-zero vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle\vec{v}\in V} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(\vec{v})=\lambda\vec{v}\ .}

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle\vec{v}} is called the eigenvector of $A$ associated with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} .