Eigenvectors and Eigenvalues

An eigenvector is a special type of vector that, when transformed by a square matrix, is scaled but remains on the same line. The vector may either point in the same direction or flip to point in the opposite direction.

The value by which the vector is scaled is called the eigenvalue. When the eigenvalue is negative, the eigenvector “flips” to point in the opposite direction.

\[A \vec{v} = \lambda \vec{v}\]

Where:

• \(A\) is the square matrix representing the transformation.
• \(\vec{v}\) is the eigenvector (a non-zero vector).
• \(\lambda\) (lambda) is the eigenvalue (a scalar value).

The eigenvalue \(\lambda\) tells you how the eigenvector is scaled:

• If \(\lambda > 1\), the vector is stretched.
• If \(0 < \lambda < 1\), the vector is shrunk.
• If \(\lambda < 0\), the vector is flipped to the opposite direction.
• If \(\lambda = 0\), the vector is collapsed to the origin (it becomes the zero vector).