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Eigenvalue Calculator

Eigenvalue Calculator

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An online eigenvalue calculator can determine the eigenvalues of the given square matrix with the characteristic equation. This eigenvalue finder allows you to substitute any matrix from 2 x 2, 3 x 3, 4 x 4, and 5 x 5. In this context, you can learn how to find eigenvalues of a matrix and much more.

What are Eigenvalues of a Matrix?

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 for matrix equations. “Eigen” is a German word that means “characteristic” or “proper”.  In short, the eigenvalue is a scalar used to transform the eigenvector.

Calculating the Trace and Determinant:

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

How to Find Eigenvalues?

The equation expression of the basic relationship between the eigenvalues and its eigenvector is Xv = λv, where λ is a scalar, X is a matrix with m rows and m columns, and v is a vector of columns. And in this relation, the true value of λ is the eigenvalue. It must satisfy the equation in order for something to have true value.

The above equation Xv = λv can be converted to X – I = 0, where “I” is the identity matrix. You can start to carry out the matrix subtraction and multiplication operations, the result will be a polynomial. The polynomial is set to zero. Then, the roots of these members are the eigenvalues. The eigenvalues ​​can be real or complex. The complex eigenvalue has a real component and an imaginary eigenvalue. If we want to find the relevant eigenvectors with an eigen value calculator that uses the original equation Xv = λv, then we calculate for every possible value of v. The value we find for v is the eigenvector.

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

Example:

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

Solution:

Finding eigenvalues for 2 x 2 matrix:

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

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

Determinant of the obtained matrix

λ^2 – 9.0 λ + 10. 0

The eigenvalue solver evaluate 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)

How to find eigenvalues of a 3×3 matrix?

  • 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.

How Eigenvalue Calculator Works?

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

Input:

  • 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 on the generate matrix button. And remove all values by clear all fields.
  • Hit the calculate button for the next procedure.

Output:

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

FAQ:

Why do we need to evaluate the eigenvalues?

Eigenvectors make it easier to understand linear transformations. These are the “axes” where the linear transformation works only by “stretching/shrinking” and/or “flipping”; the eigenvalues indicate the factor that this compression occurs.

Can the eigenvalues be zero?

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.

Where do we use eigenvalues?

We can use the eigenvalues  for:

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

The Bottom Line:

Use this eigenvalue calculator that determines the eigenvalues of the given matrix and compute the roots of the characteristic equation. The characteristic equation is obtained by equating the polynomial to zero.

Reference: 

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

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

From the source of Libre Text: Definition of eigen value, characteristic polynomial, characteristic equation, Finding Eigen Values, A Zero Eigen value, Special Types of Matrices.