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