Some [[Real Numbers|Real Number]] $\lambda$ is an [[Eigenvalue]] of a [[Matrix]] $A$ if the [[Determinant]] of $A$ minus the [[Identity Matrix]] is 0. The [[Set]] of all [[Eigenvalue|Eigenvalues]] of $A$. $\huge \lambda_{A}=\setbuild{ \lambda\in\R\,}{\, \det(A-\lambda I) =0} $ Solving for all [[Eigenvalue|Eigenvalues]] $\lambda$ for a $n\times n$ [[Matrix]] with the equation $\det(A-\lambda I)=0$ will lead to the [[Characteristic Polynomial]] of $A$ of degree $n$. If the [[Roots]] of the [[Characteristic Polynomial]] repeat, the convention is that you have multiple [[Eigenvalue|Eigenvalues]] with the same number. >[!info] Useful fact The [[Trace]] of a [[Matrix]] will always be the sum of its [[Eigenvalue|Eigenvalues]]. gonna use the version with $\Lambda$ because it looks cooler $\huge \mathrm{tr}(A) = \lambda_{1}+\lambda_{2} +\dots+\lambda_{n} $ >[!example] >Solve for all [[Eigenvalue|Eigenvalues]]: >$ >A=\mat{1&-1\\-6&0} >$ >>[!check]- Solution >> $T_{A} : \R^2 \to \R^2 \therefore$ the [[Characteristic Polynomial]] has degree $2$. >> >>$ >>\begin{align} >>\det(A-\lambda I) &= 0\\ >> >>\det\pa{ >>\mat{( >>1-\lambda & -1\\ >>-6 & -\lambda >>} >>} &= 0\\ >> >>(1-\lambda)(-\lambda) - (-6)(-1) &= 0 \\ >>\lambda^2 - \lambda - 6 &= 0 >>\end{align} >>$ >[!example] >Solve for all [[Eigenvalue|Eigenvalues]] >$ >A = \mat{ >4&0&1&1\\ >0&1&3&0\\ >0&0&-1&1\\ >0&0&0&2 >} >$ >>[!check]- Solution >>$ \begin{align} >> >>0&=\det(A-\lambda I) \\ >>&= \mat{ >>4-\lambda &0&1&1\\ >>0&1-\lambda&3&0\\ >>0&0&-1-\lambda&1\\ >>0&0&0&2-\lambda >>}\\ >>&= (4-\lambda)(1-\lambda)(-1-\lambda)(2-\lambda) \\ >>\lambda &= \set{ >>4,1,-1,2 >>} >>\end{align} >>$ ### [[Complex Numbers|Complex]] [[Eigenvalue|Eigenvalues]] Any [[Matrix]] $A$ with [[Real Numbers|Real Number]] values that has a [[Complex Numbers|Complex]] [[Eigenvalue]], with always come in [[Complex Conjugate]]. In the [[Set]] of all [[Eigenvalue|Eigenvalues]] of a [[Matrix]], you will never find a [[Complex Numbers|Complex]] [[Eigenvalue]] without its [[Complex Conjugate]]. A [[Complex Numbers|Complex]] [[Eigenvalue]] appears when the [[Function|Transformation]] has no [[Trivial|nontrivial]] fixed points, such as [[Rotation Transformation|Rotation]]. The addition of $i$ can be thought of forcing in a way of rotation with [[Eigenvalue|Eigenvalues]]/[[Eigenvector|Eigenvectors]]. >[!example] [[Eigenvalue|Eigenvalues]] of a 90 degree [[Rotation Transformation|Rotation Matrix]] >$\large \begin{align} >A&=\mat{0&-1\\1&0} \\ >\det(A-\lambda I) &= 0\\ >\sim \lambda^2 +1 &= 0\\ >\lambda &= \pm i >\end{align} $