Eigenvalues of a Matrix

Eigenvalues are numbers that characterize a matrix. These numbers are important because, associated with their eigenvectors, they make it possible to express the matrix in a simplified form, which facilitates the calculations.

for any square matrix $ M $ of size $ m \times m $ (2x2, 3x3, 4x4, etc.), eigenvalues are generally called lambda $ \lambda $ and associated with an eigenvector $ v $ if $$ M.v = \lambda v \iff (M-\lambda I_m).v = 0 $$ with $ I_m $ the identity matrix (of size $ m $).

Practically, the eigenvalues of $ M $ are the roots of its characteristic polynomial $ P $.

An eigenvalue of a matrix is always associated with an eigenvector. Use the eigenvectors calculator proposed by dCode.

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

Example: The 2x2 matrix $ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $ has for characteristic polynomial $ P(M) = x^2 − 4x − 5 = (x+1)(x-5) $. The roots of $ P $ are found by the calculation $ P(M)=0 \iff x= -1 \mbox{ or } x = 5 $. The eigenvalues of the matrix $ M $ are $ -1 $ and $ 5 $.

NB: The eigenvectors associated are $ \begin{bmatrix} 1 \\ 2 \end{bmatrix} $ for $ 5 $ and $ \begin{bmatrix} -1 \\ 1 \end{bmatrix} $ for $ -1 $

If the roots of the characteristic polynomial do not have values on $ \mathbb{R} $ then they are calculated on $ \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 as German word which means proper, characteristic.