Eigenvalues and Matrix Transformations: The Geometric Picture
A matrix is a transformation of the plane. Its eigenvalues and eigenvectors describe the directions it only stretches, compresses or flips. This guide builds that picture and connects it to the visualizer.
- Stretching and compression
- Reflection
- Eigenvector directions
- Positive and negative eigenvalues
- Repeated eigenvalues
- No real eigenvectors
- The visualizer
Stretching and compression
A=\begin{bmatrix}2&0\\0&\tfrac12\end{bmatrix} stretches the x-direction by 2 and compresses the y-direction to half. The axes are the eigenvector directions, with eigenvalues 2 and \tfrac12. Rule of thumb: |\lambda|>1 stretches, 0<|\lambda|<1 compresses.
Reflection
Swapping coordinates, A=\begin{bmatrix}0&1\\1&0\end{bmatrix}, reflects across the line y=x. Vectors along y=x do not move (\lambda=1, eigenvector (1,1)), and vectors along y=-x are flipped (\lambda=-1, eigenvector (1,-1)).
Eigenvector directions
Eigenvector directions are the lines through the origin that the matrix maps to themselves. If you apply the matrix again and again, other vectors are pulled steadily toward the direction with the largest |\lambda|, which is why that direction controls long-term behavior. See the eigenvectors guide for how to compute them.
Positive and negative eigenvalues
- \lambda>0: the direction keeps its orientation and is scaled by \lambda.
- \lambda<0: the vector is flipped through the origin, then scaled by |\lambda|.
- \lambda=1: every vector on the line stays put.
- \lambda=0: the line collapses to the origin, so the matrix is singular.
Since \det A=\lambda_1\lambda_2, a negative determinant means the transformation includes a flip.
Repeated eigenvalues
- A=\lambda I (for example 3I): every non-zero vector is an eigenvector. The matrix scales the whole plane uniformly.
- Shear \begin{bmatrix}1&1\\0&1\end{bmatrix}: \lambda=1 is repeated, but only the x-axis stays on its line. Every other vector is slid sideways. This is the defective case from the diagonalization guide.
Matrices with no real eigenvectors
A 90° rotation \begin{bmatrix}0&-1\\1&0\end{bmatrix} turns every line to a different line, so no real direction survives. Algebraically, \lambda^2+1=0 gives \lambda=\pm i, which are complex. In general, complex eigenvalues a\pm bi signal rotation combined with scaling by \sqrt{a^2+b^2}. Details of the algebra are in the eigenvalues guide.
Connection to the interactive visualizer
The 2×2 visualization in the Eigen Vector tool shows exactly these cases on a transformed grid.
- Open the Eigen Vector tool and keep the size at 2×2.
- Pick an example: Distinct real, Repeated (diagonalizable), Repeated (one eigenvector) or Complex (rotation), or type the stretch and reflection matrices above.
- Press Calculate, then Play. Dashed arrows are eigenvectors \mathbf v and the moving arrow is A\mathbf v; it stays on the same line.
- For the rotation example no eigenvector is drawn, because none is real.