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Eigenvectors and Eigenvalues

Most vectors get knocked off their line when a matrix acts. Eigenvectors do not: Av = λv. Sweep a vector around the circle and catch the moments it lines up.

Step 1 of 54

Speed

A = [(2, 1), (1, 2)] (rows). Put v on the unit circle and look at Av. The faint ellipse is where every unit vector lands. Most vectors get knocked off their own line — sweep v around and watch for the ones that are not.

A =

A\mathbf{v} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}

The dashed violet ellipse is where the whole unit circle lands. Eigenvectors are the directions where the circle and its image line up through the origin.

Eigenvectors and Eigenvalues

Most vectors get knocked off their line when a matrix acts. Eigenvectors do not: Av = λv. Sweep a vector around the circle and catch the moments it lines up.

Step 1 of 54

Speed

A =

A\mathbf{v} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}

The dashed violet ellipse is where the whole unit circle lands. Eigenvectors are the directions where the circle and its image line up through the origin.

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Eigenvectors and Eigenvalues

See also

Eigenvectors and Eigenvalues

See also