Eigenvalues and Eigenvectors

If $Ax=x$, $x$ is an Eigenvector and can be found using the Eigenvalues $\lambda$ by solving $A-\lambda I=0$, which in turn are found through finding the solving values of $\lambda$ from the equality $\det (A-\lambda I)=0$. Eigenvalues $\lambda$ denote the linear multiple of $x$ that is produced by the operation $Ax$.

$$\begin{array}{ccc}\det (A-\lambda I)=0& \text{solve for all λ}& (1)\\ A-\lambda I=0& \text{solve vor Eigenvectors x}& (2)\end{array}$$

Properties

$M=\left[\begin{array}{ccccccc}a\_11& \dots & a\_1n⋮& a\_ii& ⋮a\_n1& \cdots & a\_in\end{array}\right]$

• The Trace The sum of all values along $M$s main diagonal is exactly equal the sum of all Eigenvalues $a\_11+\cdots +a\_in=\lambda \_1+\cdots +\lambda \_n$
• The Determinant Product The product of all eigenvalues of $M$ equals the determinant of $M$. $\det (M)=\lambda \_1\dots \lambda \_n$.

Laws

• When $A$ has Eigenvalues $\lambda$, $A^n$ has Eigenvalues ${\lambda }^n$
• $\lambda =0$ is the eigenvalue of every singular matrix (It is every special solution $x\_s$)
• When $A$ is $n\times n$, $\det (A-\lambda I)$ is of rank $n$. There exists an amount of $n\ge \lambda$ Eigenvalues