Eigenvalues and eigenvectors: Difference between revisions

Revision as of 07:24, 20 November 2013

Revision as of 13:22, 23 November 2013

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:<math>A \begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 1\\0\\0 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1\\0\\0 \end{bmatrix},\quad\quad</math>

:<math>A \begin{bmatrix} 0\\0\\1 \end{bmatrix} = \begin{bmatrix} 0\\0\\3 \end{bmatrix} = 3 \cdot \begin{bmatrix} 0\\0\\1 \end{bmatrix},\quad\quad</math>

Therefore, the vectors <math>[1,0,0]^\mathsf{T}</math>, and <math>[0,0,1]^\mathsf{T}</math> and <math>[1,1,0]^\mathsf{T}</math> are eigenvectors of <math>A</math> corresponding to the eigenvalues 1, 3, and 2,3 respectively. (Here the symbol <math>{}^\mathsf{T}</math> indicates [[transpose of a matrix|matrix transposition]], in this case turning the row vectors into column vectors.)

:<math>A \begin{bmatrix} 1\\1\\0 \end{bmatrix} = \begin{bmatrix} 2\\2\\0 \end{bmatrix} = 2 \cdot \begin{bmatrix} 1\\1\\0 \end{bmatrix}.</math>

Therefore, the vectors <math>[1,0,0]^\mathsf{T}</math>, <math>[0,0,1]^\mathsf{T}</math> and <math>[1,1,0]^\mathsf{T}</math> are eigenvectors of <math>A</math> corresponding to the eigenvalues 1, 3, and 2, respectively. (Here the symbol <math>{}^\mathsf{T}</math> indicates [[transpose of a matrix|matrix transposition]], in this case turning the row vectors into column vectors.)

====Trivial cases====