Calculate the eigenvectors for a 2x2 matrix and visualize the stationary directions.
Matrix Elements
Eigenvectors
[1.00, 1.00]
Vector vβ
Eigenvectors are directions that only get scaled, not rotated, by the transformation.
Vector vβ
[1.00, -1.00]
Ξ»β3.00
Ξ»β1.00
About this calculator
Understanding Eigenvectors
When a linear transformation (represented by a matrix) is applied to space, most vectors change their direction. However, some special vectors stay on the same line they started on. These are eigenvectors. The scaling factor by which they are stretched or shrunk along that line is the eigenvalue.
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Pro Tips
Eigenvectors are not unique; any scalar multiple of an eigenvector is also an eigenvector.
If eigenvalues are complex, the eigenvectors will also have complex components, representing a rotation.
A symmetric matrix always has real eigenvalues and orthogonal eigenvectors.
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Fun Facts
"In physics, eigenvectors represent the 'principal axes' of an object, like how it naturally wants to rotate."
"Face recognition software (like Eigenfaces) uses eigenvectors of image data to identify key features."
"The resonance of a bridge is determined by the eigenvectors of its stiffness and mass matrices."
