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Eigenvalues Calculator

Name: Eigenvalues Calculator
Author: AskMathAI

Calculate matrix eigenvalues step-by-step with our advanced linear algebra calculator. Perfect for finding $\lambda$ values for any matrix size with detailed characteristic polynomial solutions.

Eigenvalues Calculator

Enter a matrix and calculate its eigenvalues with step-by-step solutions

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Master Eigenvalues with Our Advanced Calculator

Our eigenvalues calculator is designed to help students, teachers, and professionals calculate matrix eigenvalues efficiently. Whether you're working on linear algebra homework, preparing foruniversity mathematics exams, or tackling engineering problems, this tool provides comprehensive step-by-step solutions that enhance your understanding of matrix operations.

The eigenvalues calculator computes the eigenvalues of matrices by solving the characteristic equation $\det(A - \lambda I) = 0$ . Our linear algebra calculator is particularly useful for university linear algebra courses and engineering applications, where eigenvalues are used in diagonalization, stability analysis, principal component analysis, and solving differential equations.

Perfect for high school linear algebra students learning matrix properties, university studentsin advanced mathematics courses, engineering students applying eigenvalues to real-world problems, andprofessionals who need quick mathematical solutions. The eigenvalues calculator provides not just answers, but detailed explanations that help you understand the underlying concepts and improve your mathematical skills.

Eigenvalues Properties and Applications

Property	Formula	Description	Application
Characteristic Equation	$\det(A - \lambda I) = 0$	Fundamental equation for eigenvalues	Finding eigenvalues
Trace and Determinant	$\lambda_1 + \lambda_2 = \text{tr}(A)$ , $\lambda_1\lambda_2 = \det(A)$	Sum and product of eigenvalues	Verification and shortcuts
Diagonal Matrix	$\lambda_i = a_{ii}$	Eigenvalues are diagonal elements	Simple eigenvalue calculation
Similarity	$B = P^{-1}AP \Rightarrow \lambda_B = \lambda_A$	Similar matrices have same eigenvalues	Matrix transformations
Power Matrix	$A^n$ has eigenvalues $\lambda_i^n$	Eigenvalues of matrix powers	Iterative processes

Common Mistakes to Avoid

Characteristic Equation

Remember: $\det(A - \lambda I) = 0$ , not $\det(A - \lambda) = 0$ . You must subtract $\lambda$ from the diagonal elements only.

Complex Eigenvalues

Eigenvalues can be complex numbers. Don't assume they're always real. Check the discriminant of the characteristic equation.

Multiplicity

Eigenvalues can have multiplicity greater than 1. Count repeated roots properly in the characteristic equation.

How to Calculate Eigenvalues

Eigenvalue calculation involves solving the characteristic equation by finding the roots of the characteristic polynomial.

$\det(A - \lambda I) = 0$

This is the fundamental characteristic equation. The eigenvalues $\lambda$ are the values that make this determinant equal to zero.

Calculation Steps

Form A - λI

Subtract λ from diagonal elements

Calculate Determinant

Find det(A - λI)

Set to Zero

Solve det(A - λI) = 0

Find Roots

Solve the characteristic equation

Key Eigenvalue Properties

$\sum \lambda_i = \text{tr}(A)$

$\prod \lambda_i = \det(A)$

$A^n \text{ has eigenvalues } \lambda_i^n$

Examples

2×2 Matrix

$A = [[3, 1], [0, 2]]$

Steps:

Form A - λI
Calculate determinant
Solve characteristic equation

λ₁ = 3, λ₂ = 2

Symmetric Matrix

$A = [[2, 1], [1, 2]]$

Steps:

Form A - λI
Calculate determinant
Use quadratic formula

λ₁ = 3, λ₂ = 1

Diagonal Matrix

$A = [[4, 0], [0, -1]]$

Steps:

Diagonal elements are eigenvalues
No calculation needed

λ₁ = 4, λ₂ = -1

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Frequently Asked Questions

What are eigenvalues?

Eigenvalues are special scalar values $\lambda$ for which there exists a non-zero vector $\mathbf{v}$ such that $A\mathbf{v} = \lambda\mathbf{v}$ . They represent the scaling factors in the direction of their corresponding eigenvectors.

How do I calculate eigenvalues?

To calculate eigenvalues, solve the characteristic equation $\det(A - \lambda I) = 0$ . This involves forming the matrix $A - \lambda I$ , calculating its determinant, and finding the values of $\lambda$ that make the determinant zero.

Can eigenvalues be complex?

Yes, eigenvalues can be complex numbers. This happens when the characteristic equation has complex roots. Complex eigenvalues often occur in matrices with certain symmetry properties or in systems with oscillatory behavior.

What is the relationship between eigenvalues and trace/determinant?

For an $n \times n$ matrix, the sum of eigenvalues equals the trace: $\sum_{i=1}^n \lambda_i = \text{tr}(A)$ . The product of eigenvalues equals the determinant: $\prod_{i=1}^n \lambda_i = \det(A)$ .

What are repeated eigenvalues?

Repeated eigenvalues occur when the characteristic equation has multiple roots of the same value. The multiplicity of an eigenvalue is the number of times it appears as a root. This affects the number of linearly independent eigenvectors.

Is this calculator free to use?

Yes, our eigenvalues calculator is completely free to use with no limitations. You can calculate eigenvalues of as many matrices as you need.

Eigenvalues Calculator