Linear Algebra

Study of vectors, matrices, and linear transformations.

Vectors and Their OperationsMatrices and Matrix ArithmeticLinear Transformations
Determinants and InversesEigenvalues and EigenvectorsVector Spaces and Subspaces
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Vectors and Their Op...Matrices and Matrix ...Linear Transformatio...Determinants and Inv...Eigenvalues and Eige...Vector Spaces and Su...Computer Graphics an...Data Science and Mac...Robotics and Enginee...Breaking Down Word P...Checking Your Soluti...Visualizing Problems...
Advanced Topics

Eigenvalues and Eigenvectors

Special Directions and Stretching

Eigenvalues and eigenvectors are special because they reveal the secret "axes" along which a matrix transformation stretches or shrinks space.

  • An eigenvector is a vector that doesn't change direction when transformed by a matrix.
  • The eigenvalue is how much that eigenvector is stretched or squished.

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

Here, \( \vec{v} \) is the eigenvector and \( \lambda \) is the eigenvalue.

Why Are They Useful?

They're used in everything from Google's search algorithm to facial recognition and more!

Practice

  • Find the eigenvalues of the matrix \[ \begin{bmatrix} 2 & 0 \ 0 & 3 \end{bmatrix} \]
  • Discover the eigenvectors for a simple scaling matrix.

Examples

  • A scaling matrix has eigenvectors along the axes.

  • Diagonal matrices make finding eigenvalues easy!

In a Nutshell

Eigenvalues and eigenvectors show how matrices stretch or shrink certain special directions.

Next
Vector Spaces and Subspaces