# Characteristic Polynomial and Eigenvalues

For real, square matrices, the characteristic polynomial and the eigenvalues are determined here.

The matrix eigenvalue problem for a matrix **A** is given by:

**A** **x** = λ **x**.

The characteristic polynomial of **A** is determined using the determinant

det (**A** - λ **I**).

In it **I** is the identity matrix.

The characteristic polynomial is a polynomial in λ of degree n if **A** has order n×n.

The roots of this characteristic polynomial of the matrix **A** are the *eigenvalues* λ_{i} of the matrix **A**.

The characteristic polynomial is created here and all its *real* and *complex* zeros are determined.

The characteristic polynomial is created using the algorithm of *Faddejew-Leverrier*.

For *symmetrical* matrices the eigenvalues are always all *real*.

With non-symmetrical matrices *complex* eigenvalues can also occur.

These then exist as conjugate complex pairs.

## Explanation of the graphical representation

The eigenvalues of the matrix are shown as black dots in the complex number plane
and additional red circles that provide a rough estimate of the position of the eigenvalues.

This estimate is unnecessary for *triangular matrices*.

In triangular matrices, the eigenvalues are directly visible on the main diagonal.

According to the eigenvalue estimation according to *Gerschgorin*, there are circular disks in the complex number plane,

in whose union all eigenvalues of a matrix lie.

The circle centers are the diagonal elements of the matrix.

The radii of the circles are determined from the sum of the amounts of the remaining row elements.

Alternatively, you can also add up the amounts of the remaining column elements.

## Calculation example with calculation method

For the matrix

⌈ 3 3 ⌉

⌊ 1 5 ⌋

is the determinant for determining the characteristic polynomial
| 3-λ 3 |

| 1 5-λ |

Evaluating the determinant gives:

(3 - λ)·(5 - λ) - 3.

If you multiply this out, you get the characteristic polynomial of the matrix:

λ^{2} - 8λ + 12.

The characteristic polynomial has zeros

λ_{1,2} = 4 ± 2.

The two eigenvalues of the matrix thus are

λ_{1} = 2

λ_{2} = 6

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