Eigenvalues and Eigenvectors

Definition

In linear algebra, an eigenvector of a matrix is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by $\lambda$, is the factor by which the eigenvector is scaled. That is, given some eigenvector $v_{i}$ of a square matrix $\mathbf {A}$, $\mathbf {A} v_{i}=\lambda _{i}v_{i}$, where $\lambda _{i}$ is the corresponding eigenvalue of $v_{i}$. For example:

Let $\mathbf {A} ={\begin{bmatrix}3&4&-2\\1&4&-1\\2&6&-1\end{bmatrix}}$, $v_{1}={\begin{bmatrix}1\\1\\2\end{bmatrix}}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A}v_1 = \begin{bmatrix} 3 & 4 & -2\\ 1 & 4 & -1\\ 2 & 6 & -1 \end{bmatrix} \begin{bmatrix} 1\\ 1\\ 2 \end{bmatrix} = \begin{bmatrix} 3\\ 3\\ 6 \end{bmatrix} = 3\begin{bmatrix} 1\\ 1\\ 2 \end{bmatrix} = 3v_1}

Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1 } is an eigenvector of matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} } , and its corresponding eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_1 = 3 } .

Resources

• Eigenvalues and Eigenvectors, MIT Math Department
• Eigenvalues and Eigenvectors, Wikipedia
• Solving for Eigenvalues of 2x2 Matrix, Khan Academy
• Eigenvalues of a 3x3 Matrix, Khan Academy
• Finding Eigenvalues and Eigenvectors: 2x2 Matrix Example, patrickJMT
• Eigenvalues and Eigenvectors of a 3x3 Matrix,