Changes

{{merge|eigenvector}}

{{merge|eigenvector}}

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

An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a [[complex number]] <math>\lambda</math> such that

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

:<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]].

for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}</math> known as a [[eigenvector]]. The eigenvalue is the zero of a matrix's [[charateristic 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>.

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>.

[[category:Linear algebra]]

[[category:Linear algebra]]