Eigenvalues of a Matrix

The eigenvalues of a square matrix $ M $ of size $ n \times n $ (2x2, 3x3, 4x4, etc.) are scalars denoted lambda $ \lambda $ such that there exists a non-zero vector $ \vec{v} $ satisfying $ M \dot \vec{v} = \lambda \vec{v} $

The vector $ \vec{v} $ is then called the eigenvector associated with the eigenvalue $ \lambda $.

This relationship means that the linear transformation represented by the matrix $ M $ transforms the vector $ \vec{v} $ by a change of scale (multiplication by $ \lambda $) without changing its direction.

To determine the eigenvalues of a matrix $ M $, calculate the roots of its characteristic polynomial by solving the equation $ \det(M - \lambda I_n) = 0 $. The solutions $ \lambda $ of this equation are the eigenvalues of the matrix.

A square matrix of size $ n \times n $ has exactly $ n $ eigenvalues in the set of complex numbers $ \mathbb{C} $, counting multiplicities.

This property comes from the fact that the characteristic polynomial is a polynomial of degree $ n $, so it has $ n $ roots in $ \mathbb{C} $ (fundamental theorem of algebra).

Eigenvalues are numbers that characterize the behavior of a matrix when it acts on vectors. Combined with eigenvectors, they allow us to identify the specific directions in which the linear transformation represented by the matrix behaves simply like multiplication by a scalar.

This information allows us to:

— simplify a matrix by diagonalization

— more easily calculate the powers of a matrix

— study the stability of dynamical systems

— analyze data (for example, in certain dimensionality reduction methods)

In a basis of eigenvectors, the matrix can sometimes be written in a simplified form (see matrix diagonalization), which greatly simplifies calculations.

To show that a value λ is an eigenvalue of a matrix $ M $, verify that there exists a non-zero vector $ \vec{x} $ such that $ M \dot \vec{x} = \lambda \dot \vec{x} $. If this system has a solution for non-zero $ \vec{x} $, then $ \lambda $ is an eigenvalue of the matrix $ M $.

Eigenvalues are the roots of the characteristic polynomial of a matrix. If this polynomial does not have all its roots in the set of real numbers $ \mathbb{R} $, some roots then belong to the set of complex numbers $ \mathbb{C} $. Thus, a matrix whose coefficients are all real can nevertheless have complex eigenvalues.

The term 'own value' comes from the German word eigen, which means own in the sense of characteristic or specific.