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The eigenvalue calculator finds the eigenvalues of the given square matrix with the characteristic equation of polynomials along with the detailed solution.

In mathematics, eigenvalues are scalar values that are associated with linear equations (also called matrix equations). It is also called latent roots. Eigenvalues are a special set of scalars assigned to linear equations. It is mainly used in matrix equations. “Eigen” is a German word that means “characteristic” or “proper”. In short, the eigenvalue is a scalar used to transform the eigenvector.

For a 2×2 matrix, the trace and the determinant of the matrix are useful to obtain two very special numbers to find the eigenvectors and eigenvalues. Fortunately, the eigenvalue calculator will find them automatically. If you want to check whether the correct answer is given or just want to calculate it manually, then please do the following:

**Trace:** The trace of the matrix is defined as the sum of the elements on the main diagonal (from the top left to bottom right). It is also equal to the sum of eigenvalues (counted with multiplicity). In the case of a 2×2 matrix,

Tr X = x_1 + b_2

**Determinant:** The matrix determinant is useful in several additional operations, such as finding the inverse of the matrix. For the 2×2 matrix,

|X| = x_1 y_2 – x_2 y_1

However, an Online Jacobian Calculator helps you to find the Jacobian matrix and the determinant of the set of functions.

Calculate eigenvalues for the matrix {{6,1}, {8, 3}}.

**Solution: **

Finding eigenvalues for 2 x 2 matrix:

First, the eigenvalues calculator subtracts λ from the diagonal entries of the given matrix

$$ \begin{vmatrix} 6.0 – λ \\ 1.0 && 8.0 \\ 3.0 – λ \end{vmatrix} $$

The determinant of the obtained matrix

λ^2 – 9.0 λ + 10. 0

The eigenvalue solver evaluates the equation

λ^2 – 9.0 λ + 10. 0 = 0

Roots (Eigen Values)

λ_1 = 7.7015

λ_2 = 1.2984

(λ_1, λ_2) = (7. 7016, 1. 2984)

The online calculator solves the eigenvalues of the matrix by computing the characteristic equation by following these steps:

- First, select the size for the matrix from the drop-down list.
- Now, substitute the values in all fields. You can generate random values for the matrix by clicking the generate matrix button. Remove all values by clearing all fields.
- Hit the calculate button for the next procedure.

- The matrix eigenvalue calculator displays the values and solves the equation.
- It also takes the determinant of the obtained matrix and provides root values.

- To find the eigenvalues of a 3×3 matrix, X, you need to:
- First, subtract λ from the main diagonal of X to get X – λI.
- Now, write the determinant of the square matrix, which is X – λI.
- Then, solve the equation, which is the det(X – λI) = 0, for λ. The solutions of the eigenvalue equation are the eigenvalues of X.

The eigenvalues can be zero. We do not treat zero vectors as eigenvectors: since X 0 = 0 = λ0 for each scalar λ, the corresponding eigenvalue is undefined.

We can use the eigenvalues for:

- Eigenvalue analysis is used in the design of autostereoscopic systems to reproduce car vibrations caused by music.
- Electrical engineering: the application of eigenvalues can be used to separate three-phase systems by converting symmetrical components.

From the source Wikipedia: Characteristic value, the characteristic polynomial, Eigenvalues of matrics, Algebraic multiplicity, Eigenspaces, geometric multiplicity, and the eigenbasis for matrices.

From the source of Medium: Eigenvalues uses, building blocks of Eigenvalues, Matrix Addition, Multiplying Scalar With A Matrix, Matrices Multiplication.