Eigenvalues of a Matrix

Consider \( M \) a square matrix of size \( m \times m \), the eigenvalues of \( M \) are the roots of the characteristic polynomial \( P \) of the matrix \( M \).

Eigenvalues are generally called \( \lambda \) 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 \( m \)).

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

Example: $$ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \Rightarrow P(M) = x^2 − 4x − 5 = (x+1)(x-5) $$

Example: $$ P(M)=0 \iff x= -1 \mbox{ or } x = 5 $$ The eigenvalues of the matrix \( M \) are \( -1 \) and \( 5 \). And the eigenvector associated are \( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) for \( 5 \) and \( \begin{bmatrix} -1 \\ 1 \end{bmatrix} \) for \( -1 \).

Normally the definition of the eigenvector exclude the zero vector. However, if there are not as many independent eigenvectors as eigenvalues, dCode will display a null vector.