Eigenvalue Calculator

An eigenvalue calculator is a dynamic tool that displays the eigenvalue for a given matrix. Our eigenvalue finder is designed with user needs in mind to provide precise and fast calculations.

Steps to Use This Eigen Value Calculator:

Select the order of matrix (Our tool supports maximum 5 x 5 matrix)
Input all the entities of the matrix
Click ‘Calculate’

Applications of Eigenvalue Calculator In Mathematics:

Our calculator provides assistance in multiple domains of mathematics, some of which include:

Diagonalization of Matrices

With eigenvalues, you can check if a matrix can be diagonalized or not. This is because a diagonal matrix reduces the exponents or powers of the matrix, which further helps in resolving systems of linear equations or differential equations.

Solving Differential Equations

You can use the eigenvalues to determine the behaviour of solutions for linear differential equations. For example, in dynamical systems, eigenvalues determine stability of equilibria.

Linear Transformations & Geometry

Eigenvalues clearly describe how a transformation scales Eigenvectors along specific directions.

Spectral Theorem & Quadratic Forms

Determining eigenvalues of symmetric or Hermitian matrices provide critical insights into quadratic forms.

Characteristic Polynomial

Eigenvalues are the roots of the characteristic polynomial of a matrix. This relationship is used to study algebraic properties of matrices.

Stability Analysis in Dynamical Systems

In discrete or continuous systems, eigenvalues of the system matrix determine stability:

Negative real parts → stable.
Positive real parts → unstable.
Purely imaginary → oscillatory.

Concept of Eigenvalue in a Matrix

An eigenvalue is a special scalar (λ) associated with a square matrix. It tells us how much an eigenvector is stretched, compressed, or flipped when the matrix is applied as a linear transformation.

Formally, if A is a square matrix and v is a non-zero vector, then λ is an eigenvalue if:

Av = λv

This means multiplying matrix A by vector v results in the same vector scaled by λ.

If λ > 0, the vector keeps its direction but changes length.
If λ < 0, the vector is flipped in direction.
If |λ| > 1, the vector is stretched.
If 0 < |λ| < 1, the vector is shrunk.

Eigenvalues only exist for square matrices (2×2, 3×3, or higher). Each eigenvalue is tied to at least one corresponding eigenvector, which is the direction that remains unchanged except for scaling.

Evaluation Formula:

The scalar value like square matrix and nonzero vectors is associated with the system of linear equations multiplied, then the result is a scaled version of V and λ. In the formulaic term, the linear transformation corresponds to the matrices given below:

$$ Av \;=\; λv $$

In order to find eigenvalues the above can be rewritten as follows:

$$ (A \;-\; λI)v \;=\; 0 $$

This formula is used by the eigenvalues calculator in which:

I indicate the identity matrix
A shows the scalar matrix
V is a nonzero vector

Example:

Suppose a special set of square matrix $$ \left[\begin{matrix}2 & 1\\2 & 3 \end{matrix}\right] $$

Solution:

Step # 1:

Subtract λ from the diagonal entries of the given matrix

$$ \left[\begin{matrix}2.0 - \lambda & 1.0\\2.0 & 3.0 - \lambda\end{matrix}\right] $$

Step # 2:

Cross-multiply the matrix values together to get the eigenvalue or directly use the matrix eigenvalue calculator 3x3.

( 2 - λ ) ( 3 - λ ) - ( 1 ) ( 2 )

6 - 2λ - 3λ + λ^2 - 2

6 - 5λ + λ^2 - 2

Step # 3:

The determinant of the obtained matrix

λ^2 - 5λ + 4 = 0

λ^2 - 4λ - λ + 4 = 0

λ ( λ - 4 ) - 1 ( λ - 4 ) = 0

( λ - 4 ) ( λ - 1)

So the roots of the Eigenvalue that is evaluated by the given matrix are as follows:

λ1 = 4
λ2 = 1

Faqs

Can Eigenvalue Be Used for Non-Square Matrices?

Eigenvalues are only associated with square matrices and this term is not used for non-square matrices.

Can Eigenvalue Be Infinitive?

No, it is not possible! If it is possible then there is a column vector.

What Are The Types of Eigenvalues?

There are the following types that are as follows:

One Eigenvalue
Two distinct real values
Complex conjugate

Can this work for non-square matrices?

No, our eigenvalue solver does not function for non-square values, as it is mandatory that the matrix must be square.

What does a zero eigenvalue indicate?

A zero eigenvalue means that the given matrix is singular and shows linear dependence.

Why do negative eigenvalues reverse direction?

Because the scalar factor λ is negative, multiplying by it flips the sign of the eigenvector, which means the direction is reversed.

What if eigenvalues are complex?

Complex eigenvalues appear in conjugate pairs for real matrices. Our tool supports complex eigenvalues but it indicates the user about the situation.

How accurate are the results?

You will get accurate results each time. However, chances are there that the tool rounds off the value for matrices with bigger entities.

Additional References:

➥ Golub, G.H. and Van der Vorst, H.A. 'Eigenvalue computation in the 20th century', Computers and Mathematics with Applications, 40(3-4), pp. 113-140. Available at: https://www.sciencedirect.com/science/article/pii/S0377042700004131

➥ Horn, R.A. and Johnson, C.R. Matrix Analysis. 2nd ed. Cambridge: Cambridge University Press. Available at: https://www.cambridge.org/core/books/matrix-analysis/9CF2CB491C9E97948B15FAD835EF9A8B

➥ For further reference, the Harvard lecture video by Gilbert Strang on eigenvalues is available at: https://www.youtube.com/watch?v=PFDu9oVAE-g

➥ Wikipedia - Eigenvalues and eigenvectors. Available at: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

➥ LibreTexts (no date) Eigenvalues and Eigenvectors. Available at: https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.01%3A_Eigenvalues_and_Eigenvectors#:~:text=An%20eigenvalue%20of%20A%20is,v%20has%20a%20nontrivial%20solution