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.

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.

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 eigenvector associated are $ \begin{bmatrix} 1 \\ 2 \end{bmatrix} $ for $ 5 $ and $ \begin{bmatrix} -1 \\ 1 \end{bmatrix} $ for $ -1 $

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