Difference between revisions of "Eigenvalue"
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(geometric multiplicity) |
(restoring some material, clarifying that the eigenvalue is simply a member of the field the vector space is built on) |
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{{merge|eigenvector}} | {{merge|eigenvector}} | ||
| − | An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a | + | 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 | + | 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]] | ||
Revision as of 21:46, May 2, 2010
It has been suggested that this article or section be merged with [[::eigenvector|eigenvector]]. (Discuss)
An eigenvalue of a square matrix is a scalar such that
for some non-zero vector 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 is called the eigenspace of . The dimension of this space is called the geometric multiplicity of the eigenvalue.
