>[!quote] In a Nutshell >Eigenvectors are the directions that are only scaled when a given [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|transformation]] is applied to it. The scaling is denoted the eigenvalues. Intuitively, computing the eigenvalues is decomposing the complex problem of a transformations into its simpler, linear components. This can be achieved by changing the basis accordingly. --- >[!info] Special Eigenvalue Problem >For a given [[Matrices|matrix]] $\mathbf{A} \in \mathbb{C}^{n\times n}$ and $n \in \mathbb{N}$, $\lambda \in \mathbb{C}$ is an eigenvalue if there is a $x \in \mathbb{C}^{n} \setminus \{0\}$, for which$\mathbf{A}x=\lambda x.$ - $x$ is eigenvector for eigenvalue $\lambda$ - Eigenvectors are not unique - The polynomial $\det(\mathbf{A-\lambda I})$ is called *characteristic polynomial* (degree $n$) - If $x$ is eigenvector for eigenvalue $\lambda$, then $p(\mathbf{A})x=p(\lambda)x$ for any polynomial $p$ - For [[Matrices|symmetric matrices]] $\mathbf{A}\in \mathbb{R}^{n \times n}$ - All eigenvalues are real, eigenvectors can be chosen as real - Eigenvectors of different eigenvalues are orthogonal - For every eigenvalue of multiplicity $m$ there are $m$ linear independent eigenvectors - There is an orthogonal basis $\{\mathbf{u}_{1}, ...,\mathbf{u}_{n} \}$ of $\mathbb{R}^n$ of eigenvectors to the eigenvalues $\lambda_1,...,\lambda_n$ $\mathbf{Au}_{j}=\lambda_{j}\mathbf{u}_{j}, \qquad \langle\mathbf{u_{j},\mathbf{u}_{i}} \rangle=\delta_{i,j}=\cases{1 \quad \text{if } i=j\\0 \quad \text{else}}$ --- #### Orthonormal Basis of Eigenvectors For every real [[Matrices|symmetric matrix]] $\mathbf{A}\in \mathbb{R}^{n \times n}$, there exists an orthonormal basis of eigenvectors to the eigenvalues $\lambda_{i}$. Collecting these eigenvectors in a matrix $\mathbf{U}$, one obtains a similarity transformation that brings $\mathbf{A}$ to diagonal form:$\mathbf{U^{T}AU}= \Lambda$ --- #### Computation Computing the eigenvalues using the characteristic polynomial is infeasible, because computation of this polynomial is prone to rounding errors and the following computation of roots is bad- [[Condition Number|conditioned]]. Therefore, use iterative algorithms, depending on the goal: - Compute spectral radius (largest eigenvalue) only - Direct [[Power Iteration|power iteration]] - Compute specific eigenvalue - Inverse [[Power Iteration|power iteration]] - Compute all eigenvalues - [[QR-Algorithm|QR-algorithm]]