4.2 Finding Eigenvalues and Eigenvectors

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Given a matrix $A$, a non-zero vector $\overrightarrow{v}$ is an eigenvector of $A$ if $A\overrightarrow{v}$ is a multiple of $\overrightarrow{v}$. In other words, $\overrightarrow{v}$ is an eigenvector of A if $A\overrightarrow{v}=\lambda \overrightarrow{v}$. Note that $\lambda$ can equal zero, but that the zero vector is not considered an eigenvector.

The polynomial $P(\lambda )=det(A-\lambda I)$ is called the characteristic polynomial. If A is an nxn matrix, this polynomial has degree n. Its roots are the eigenvalues of A.

The eigenvectors of a matrix A for a given eigenvalue $\lambda$ form a subspace (we also throw in $\overrightarrow{0}$), called the eigenspace of A for eigenvalue $\lambda$. This is written $E_{\lambda }=Nul(A-\lambda I)$.