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{{merge|eigenvector}}

{{merge|eigenvector}}

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

An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a scalar <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}^n</math> known as a [[eigenvector]]. The eigenvalue is the zero of a matrix's [[charateristic polynomial]], the degree of the corresponding root is called the '''algebraic multiplicity''' of the eigenvalue.

for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}^n</math> known as a [[eigenvector]]. The eigenvalues are the zeroes of a matrix's [[charateristic polynomial]], the degree of the corresponding root is called the '''algebraic multiplicity''' of the eigenvalue.

 

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 dimension of this space is called the '''geometric multiplicity''' of the 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 dimension of this space is called the '''geometric multiplicity''' of the eigenvalue.

[[category:Linear algebra]]

[[category:Linear algebra]]