Eigenvalue Calculator

Calculate the eigenvalues of a matrix upto size 6x6 with our free eigenvalue calculator.

Calculate Eigenvalues of a Matrix

Matrix Size

Matrix Elements

Related Calculators →

Mathematics

Null Space Calculator

Calculate the null space of a matrix upto size 10x10 with the null space calculator.

Mathematics

RREF Calculator

Calculate the reduced row echelon form of a matrix with our RREF calculator.

Mathematics

Diamond Problem Solver

Calculate the factors of a quadratic equation using the diamond method with our Diamond Problem Solver

Eigenvalue Calculator: Find all eigenvalues of a matrix

In linear algebra, eigenvalues are special scalar values associated with a square matrix or linear transformation. They represent the factor by which a corresponding non-zero vector, called an eigenvector, is scaled (stretched, shrunk, or reversed) when the linear transformation is applied. Calculating these values manually involves solving the characteristic equation, det(A - λI) = 0, where A is the matrix, λ represents the unknown eigenvalue, and I is the identity matrix. An eigenvalue calculator is a computational tool, often available online or as part of mathematical software packages, designed to automate this process. Users input the elements of a square matrix, and the calculator efficiently computes and outputs its eigenvalues, and often the corresponding eigenvectors as well.

Eigenvalue Calculator: Quick Overview

Calculate eigenvalues of matrices up to 6x6 with our popular Eigenvalue Calculator. Get step-by-step solutions and detailed explanations.

Instant Calculations

Get eigenvalues and detailed steps instantly in lesser time than manual calculation

Matrix Operations

Support for matrices up to 6x6 size, covering all basic needs

Educational Content

Learn about eigenvalues and their applications

Smart Features

AI-powered explanations and step-by-step solutions

Perfect for students, engineers, and professionals working with linear algebra. Includes detailed examples and real-world applications.

The Eigenvalue Calculator is your go-to online tool for quickly and accurately computing eigenvalues of the matrix. Supporting matrices up to 6x6, our calculator provides detailed, step-by-step solutions, making it an excellent resource for students, educators, and professionals in fields like linear algebra, physics, and engineering.

Our Eigenvalue calculator uses advanced numerical algorithms to ensure accurate results, presented in a clear and easy-to-understand format. Whether you're studying the fundamentals of linear algebra, tackling complex engineering problems, or exploring quantum mechanics, this tool simplifies the process of eigenvalue calculation and enhances your understanding of the underlying mathematical principles.

What is a Matrix?

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are fundamental in linear algebra and are used to represent linear transformations, solve systems of linear equations, and perform various other mathematical operations.

A matrix with m rows and n columns is called an m x n matrix. Each element of a matrix is identified by its row and column indices (e.g., a₁₁ is the element in the first row and first column).

Types of Matrices

Matrices play a fundamental role in mathematics, engineering, and computer science. Different types of matrices unique properties and applications, making them essential in various fields, including linear algebra, physics, and machine learning. Below is a detailed overview of the most important types of matrices, along with their definitions and examples.

1. Square Matrix

A matrix is called a square matrix if the number of rows and columns are equal. It is of size n x n

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

2. Rectangular Matrix

A matrix of size m x n, with a different number of rows and columns is called a rectangular matrix. It is of size m x n

B = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \end{bmatrix}

3. Diagonal Matrix

A diagonal matrix is a square matrix of size n x n, where all non-diagonal elements are zero.

C = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}

4. Identity Matrix

An identity matrix is a square matrix of size n x n, in which all the diagonal elements are 1, and all other elements are 0. Identity Matrix are mostly represented by letter 'I'

I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

5. Zero (Null) Matrix

A matrix of size m x n in which all elements are zero is called a zero or null matrix.

I = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

6. Symmetric Matrix

A symmetric matrix is a square matrix of size n x n, that is equal to its transpose, i.e., A = Aᵀ.

S = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 5 & 6 \\ 4 & 6 & 8 \end{bmatrix}

7. Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix of size n x n, where Aᵀ = -A.

K = \begin{bmatrix} 0 & -2 & -3 \\ 2 & 0 & -4 \\ 3 & 4 & 0 \end{bmatrix}

8. Upper Triangular Matrix

A square matrix of size n x n, where all elements below the main diagonal are zero.

U = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}

9. Lower Triangular Matrix

A square matrix of size n x n, where all elements above the main diagonal are zero.

L = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \end{bmatrix}

10. Singular Matrix

A square matrix of size n x n, whose determinant is zero, meaning it has no inverse.

S = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}

Determinant => (2 \times 2) - (4 \times 1) = 4 - 4 = 0

What is the Determinant of a Matrix?

The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. It provides important information about the matrix, such as whether the matrix is invertible (non-singular) and the volume scaling factor of the linear transformation described by the matrix.

For a 2x2 matrix, the determinant is calculated as follows:

Determinant \ of \ a \ 2x2 \ matrix: det (\begin{bmatrix} a & b \\ c & d \end{bmatrix}) = ad - bc

For larger matrices, the determinant can be calculated using methods like cofactor expansion or row reduction.

What is an Eigenvalue?

An eigenvalue is a special scalar (number) λ associated with a square matrix A that satisfies the equation Av = λv for some non-zero vector v. The vector v is called an eigenvector. Eigenvalues represent how the matrix transforms vectors in terms of scaling, and they play a crucial role in understanding linear transformations.

For a matrix A, if Av = λv for some non-zero vector v, then λ is an eigenvalue of A, and v is its corresponding eigenvector.

What is an Eigenvector?

An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, results in a vector that is a scalar multiple of itself. This scalar is the eigenvalue corresponding to that eigenvector. In other words, the direction of the eigenvector remains unchanged (or is reversed) by the linear transformation represented by the matrix.

Eigenvectors are fundamental in understanding the behavior of linear transformations and are used in various applications, including stability analysis, vibration analysis, and quantum mechanics.

How to Find Eigenvalues and Eigenvectors?

To find the eigenvalues and eigenvectors of a square matrix A, follow these steps:

Form the Characteristic Equation: Subtract λI from A, where λ is a scalar (the eigenvalue) and I is the identity matrix. Then, take the determinant of (A - λI) and set it equal to zero: det(A - λI) = 0.
Solve for Eigenvalues (λ): Solve the characteristic equation for λ. The solutions are the eigenvalues of the matrix A.
Find Eigenvectors for Each Eigenvalue: For each eigenvalue λ, substitute it back into the equation (A - λI)v = 0, where v is the eigenvector. Solve this system of linear equations to find the eigenvector(s) corresponding to that eigenvalue.

What is an Eigenvalue Calculator?

An Eigenvalue Calculator is a specialized tool that computes the eigenvalues of a given square matrix. It works by finding the roots of the characteristic polynomial, which is obtained by solving the equation det(A - λI) = 0, where A is the input matrix, λ represents the eigenvalues, and I is the identity matrix.

The formula to find eigenvalues: det(A - λI) = 0

How to Use an Eigenvalue Calculator?: Learn step-by-step

Step 1

Select the size of your matrix (2x2 up to 6x6)

Step 2

Enter the matrix elements in the provided grid

Step 3

Click Calculate to find eigenvalues

Step 4

View the characteristic polynomial and eigenvalues

Step 5

Use AI explanation for detailed understanding

Special Cases for Eigenvalue Calculator

Matrix with All Zero Entries

If A is a zero matrix, all eigenvalues are λ = 0

A = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

=> Eigenvalue (λ) = 0

Diagonal Matrix

If A is diagonal, eigenvalues are simply the diagonal elements.

A = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}

=> Eigenvalues (λ) = 3, 7

Singular Matrix (Determinant = 0)

At least one eigenvalue is λ = 0.

A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}

=> One Eigenvalue (λ) = 0

Identity Matrix

If A = I, all eigenvalues are λ = 1.

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

=> Eigenvalues (λ) = 1, 1

Symmetric Matrix

Eigenvalues are always real numbers.

A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}

=> Real Eigenvalues

Defective Matrix (Not Diagonalizable)

May have repeated eigenvalues but not enough independent eigenvectors.

A = \begin{bmatrix} 5 & 1 \\ 0 & 5 \end{bmatrix}

=> Repeated Eigenvalues (λ) = 5, 5

Complex Eigenvalues

If the matrix contains complex numbers or has complex-conjugate pairs.

A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

=> Complex Eigenvalues (λ) = i, -i

Repeated Eigenvalues (Multiplicity)

A repeated eigenvalue might indicate a defective matrix.

A = \begin{bmatrix} 6 & 1 \\ 0 & 6 \end{bmatrix}

=> Repeated Eigenvalues (λ) = 6, 6

Features of our Calxify's Eigenvalue Calculator

Accurate Calculations

Get precise eigenvalues using advanced numerical methods.

Step-by-Step Solutions

Follow the detailed calculation process from characteristic polynomial to final eigenvalues.

Matrix Size Flexibility

Handle matrices of various sizes, from 2x2 up to 6x6.

Educational Support

Learn about eigenvalues through comprehensive explanations.

AI-Powered Insights

Get intelligent explanations of calculations and their significance.

Applications of Eigenvalues

1. Principal Component Analysis (PCA)

Eigenvalues are used to identify the most important features in a dataset. The eigenvalues represent the amount of variance explained by each principal component.

Example: In image compression, PCA uses eigenvalues to reduce image dimensions while preserving important features.

2. Quantum Mechanics

Eigenvalues represent the possible values that can be measured for physical observables like energy and momentum in quantum systems.

Example: The energy levels of an electron in a hydrogen atom are eigenvalues of the Hamiltonian operator.

3. Structural Engineering

Eigenvalues help determine the natural frequencies and modes of vibration in structures.

Example: Calculating the resonant frequencies of bridges to prevent structural failure.

Example Calculation: Finding Eigenvalues

Let's determine the eigenvalues of a given 2×2 matrix using the characteristic equation method.

Given Matrix

A = \begin{bmatrix} 4 & -2 \\ 1 & 1 \end{bmatrix}

Step 1: Write the Characteristic Equation

det(A - \lambda I) = 0

Step 2: Equate with Determinant

\therefore \begin{vmatrix} 4-\lambda & -2 \\ 1 & 1-\lambda \end{vmatrix} = 0

Step 2: Expand the Determinant

\therefore (4-\lambda)(1-\lambda) - (-2)(1) = 0

Step 3: Simplify

\therefore \lambda^2 - 5\lambda + 6 = 0

Step 4: Solve the Quadratic Equation

\therefore (\lambda - 2)(\lambda - 3) = 0

Step 5: Find Eigenvalues

\lambda_1 = 3, \lambda_2 = 2

Conclusion

The given matrix has eigenvalues λ₁ = 3 and λ₂ = 2. These values indicate how the matrix transforms vectors in its space, a crucial concept in linear algebra.

Frequently Asked Questions

Q1. What are eigenvalues and eigenvectors?

•

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. Given a square matrix A, an eigenvalue (λ) is a scalar such that there exists a nonzero vector v (the eigenvector) satisfying the equation A * v = λ * v. This means that the matrix only scales the vector without changing its direction.

Q2. How do I calculate eigenvalues of a matrix?

•

To calculate eigenvalues of a matrix A, solve the characteristic equation det(A - λI) = 0, where I is the identity matrix of the same size as A. The roots of this equation are the eigenvalues. You can use the Calxify's Eigenvalue Calculator tool for quick computation

Q3. What is the characteristic polynomial, and how is it used to find eigenvalues?

•

The characteristic polynomial of a square matrix A is obtained from det(A - λI). The roots of this polynomial give the eigenvalues of the matrix.

Q4. What is an eigenvector?

•

An eigenvector of a matrix A is a nonzero vector v that satisfies A * v = λ * v, where λ is an eigenvalue of A. The vector v remains in the same direction after transformation by A.

Q5. How do I find the eigenvectors corresponding to an eigenvalue?

•

To find the eigenvectors corresponding to an eigenvalue λ, solve (A - λI)v = 0, where I is the identity matrix and v is the eigenvector.

Q6. What are eigenvalues and eigenvectors used for?

•

Eigenvalues and eigenvectors are used in various fields, including physics, engineering, computer science, and finance. Applications include stability analysis, principal component analysis (PCA), vibration analysis, and quantum mechanics.

Q7. Can a matrix have complex eigenvalues?

•

Yes, a matrix can have complex eigenvalues, especially if it has real but non-symmetric entries. The eigenvalues appear as complex conjugate pairs when the matrix has real coefficients.

Q8. Is it possible for a matrix to lack eigenvectors?

•

Yes, a matrix may not have enough linearly independent eigenvectors. This happens when the matrix is defective, meaning it does not have a complete set of eigenvectors for diagonalization.

Q9. Are all eigenvectors linearly independent?

•

Not necessarily. If a matrix has repeated eigenvalues, its eigenvectors may be linearly dependent.

Q10. What happens when a matrix lacks linearly independent eigenvectors?

•

If a matrix lacks a full set of linearly independent eigenvectors, it cannot be diagonalized. Instead, it can be transformed into Jordan normal form.

Q11. How are eigenvalues used in quantum mechanics?

•

In quantum mechanics, eigenvalues correspond to measurable quantities such as energy levels of an atom, and eigenvectors represent the possible quantum states.

Q12. Can a matrix have all three eigenvalues the same?

•

Yes, a matrix can have three identical eigenvalues. If the matrix is diagonalizable, it will have three linearly independent eigenvectors.

Q13. Are all the eigenvalues of a symmetric matrix distinct?

•

Not necessarily, but all eigenvalues of a symmetric matrix are real, and it always has an orthogonal set of eigenvectors.

Q14. How am I supposed to calculate the Eigenvalue(s) of a Matrix?

•

You need to solve det(A - λI) = 0. You can use Calxify's Eigenvalue Calculator for an easier computation.

Q15. What is eigenvalue decomposition?

•

Eigenvalue decomposition is the process of decomposing a matrix into the form A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P contains the corresponding eigenvectors.

Q16. What sizes of matrices can an eigenvalue calculator handle?

•

Our Calxify's Eigenvalue calculator can handle up to 6x6 matrices

Q17. What are the benefits of using an eigenvalue calculator?

•

It saves time, eliminates manual errors, and provides step-by-step solutions for better understanding.

Q18. Can eigenvalues be zero? What does that mean?

•

Yes, an eigenvalue can be zero. This means the matrix is singular and does not have an inverse.

Q19. Can eigenvectors be zero?

•

No, by definition, eigenvectors must be nonzero.