Eigenvalues of a Matrix

Eigenvalues for any square matrix $ M $ of size $ m \times m $ (2x2, 3x3, 4x4, etc.), are the scalar values noted with character lambda $ \lambda $ that are associated with an eigenvector $ \vec{v} $ such as $$ M.\vec{v} = \lambda \vec{v} $$

Practically, the eigenvalues $ \lambda $ of a matrix $ M $ are the roots of its characteristic polynomial $ P $ as $ (M-\lambda I_m).\vec{v} = 0 $$ (with $ I_m $ the identity matrix of size $ m $).

To determine/find the eigenvalues of a matrix, calculate the roots of its characteristic polynomial.

A square matrix of size/dimension $ n $ has $ n $ eigen values. Be careful however, certain eigen values can be identical, to know the number of distinct eigenvalues (without multiplicity) then calculate the distinct roots of the characteristic polynomial of the matrix.

Eigenvalues are numbers that characterize a matrix. These numbers are important because, associated with their eigenvectors, they make it possible to determine the eigendirections of the matrix and to express it in a basis in a simplified form (see matrix diagonalization), which facilitates calculations.

To determine that a value λ is an eigenvalue of a matrix $ M $, show that there exists a nonzero vector $ \vec{x} $ such that $ M . \vec{x} = \lambda . \vec{x} $. If this equation has a solution for $ \vec{x} $, then $ \lambda $ is an eigenvalue of the matrix $ M $.

If the roots of the characteristic polynomial do not have values on the real set $ \mathbb{R} $ then they are calculated on the complex set $ \mathbb{C} $ which introduces complex eigenvalues.

This case can occur even if the values of the matrix are all real numbers.

Eigenvalues are called eigen because it is a German word which means proper, characteristic.