Eigenvalues

An eigenvalue, often denoted with $\lambda$, is a scalar value which indicates how much an eigenvector is scaled when a matrix acts on it. They are sometimes also known as “characteristic values”.

Eigenvalue-eigenvector problems are usually illustrated by the equation: $Ax=\lambda x$

• A is a square matrix
• x is an eigenvector
• $\lambda$ is an eigenvalue

Example

$$AX=\left[\begin{array}{ccc}0& 5& -10\\ 0& 22& 16\\ 0& -9& -2\end{array}\right]\times \left[\begin{array}{c}-5\\ -4\\ 3\end{array}\right]$$ $$AX=\left[\begin{array}{c}-50\\ -40\\ 30\end{array}\right]=10\left[\begin{array}{c}-5\\ -4\\ 3\end{array}\right]$$

In the problem above, we can see that the eigenvalue is $\lambda =10$ for the matrix A.

We may also see that $\left[\begin{array}{c}-5\\ -4\\ 3\end{array}\right]$ is the eigenvector.

Characteristic Equation §

To find eigenvalues, you solve the characteristic equation:

$$det(A-\lambda I=0)$$

Where $A$ is the given matrix, $\lambda$ is the eigenvalue, and $I$ is the identity matrix of the same size as $A$.

Diagnolization §

Source(s) & Additional Reading: §

• 7.1: Eigenvalues and Eigenvectors of a Matrix, A First Course in Linear Algebra, Kuttler

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