Eigenvalue

It has been suggested that this article or section be merged with [[::eigenvector|eigenvector]]. (Discuss)

An eigenvalue of a square matrix <math>A</math> is a real number or complex number <math>\lambda</math> such that

<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math>

for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}</math> known as a eigenvector. The eigenvalues are the zeroes of a matrix's characteristic polynomial.

The product of all the eigenvalues of a matrix equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue.

The span of all the eigenvectors corresponding to a fixed eigenvalue <math>\lambda</math> is called the eigenspace <math>E_\lambda</math> of <math>A</math>.