Gershgorin circle theorem

Gerschgorin Circles used in numerical linear algebra, a branch of mathematics, for the estimation of eigenvalues ​​. With their help can be easily specified geographic areas in which the eigenvalues ​​of a matrix are and under specific conditions even how many eigenvalues ​​in these are included.

They are named after the Belarusian mathematician Semyon Aronowitsch Gerschgorin.

Definition

Let be a square matrix with entries from (ie ), then the belonging to the i-th diagonal element Gerschgorin circle is defined as follows:

Wherein the closed circle of radius referred to the point.

Since the set of eigenvalues ​​(spectrum) of the same as that of, another family of circles with the same properties can also be determined by column:

Estimation of eigenvalues

The following applies:

  • The spectrum is from a subset of
  • If there is a subset of, so that:

Or, more memorable: Each connected component of the union of all the Gerschgorin circle disks contains as many eigenvalues ​​as diagonal elements of the matrix.

By the possibility of the line as well as circuits to calculate both columns ( the eigenvalues ​​of the transposed matrix are the same ), two assessments per diagonal element can be found in non-symmetric matrices.

Examples

For the matrix

There are the following Gerschgorin circles (column- and row-wise ):

  • And the diagonal element
  • And the diagonal element
  • And the diagonal element

Since the set intersection is empty, is in exactly one eigenvalue and are exactly two.

The actual eigenvalues ​​of the matrix are rounded 1.8692, 4.8730 and 6.2578 and actually contained in the territories indicated above.

The matrix

Is symmetric and real, so all eigenvalues ​​are real and there are the following real intervals ( Gerschgorin circles):

  • The diagonal element
  • The diagonal element
  • The diagonal element

Since only the diagonal element is different from zero in the second row and column of this matrix, an eigenvalue can be determined easily, the other two lie in the intervals and, thus can be directly identified as positive definite. The actual eigenvalues ​​of the matrix are thus about 4.6972, and 8.3028 7.

Use

The Gerschgorin circles provide an easy way to determine properties of matrices in numerical analysis. Contains example no Gerschgorin circle to zero, the matrix is invertible. This property is summarized in the concept of strictly diagonally dominant matrix. Similarly, the definiteness can be roughly estimated using the Gerschgorin circles with symmetric or Hermitian matrices often.

Domain (mathematical analysis) Closed set Symmetric matrix Hermitian matrix
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