Matrix Eigenvalue and Eigenvector Calculator

Matrix Eigenvalue Problem:

\[ A\mathbf{v} = \lambda\mathbf{v} \]

where:

\( A \) is a square matrix
\( \lambda \) are eigenvalues (scalars)
\( \mathbf{v} \) are eigenvectors (vectors)

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1. What are Eigenvalues and Eigenvectors?

Eigenvalues (λ) and eigenvectors (v) are fundamental concepts in linear algebra. For a square matrix A, an eigenvector is a non-zero vector that only changes by a scalar factor when that matrix is applied to it. The corresponding eigenvalue is the scalar factor by which the eigenvector is scaled.

\[ A\mathbf{v} = \lambda\mathbf{v} \]

2. How the Calculator Works

The calculator solves the characteristic equation:

\[ \det(A - \lambda I) = 0 \]

For 2×2 matrices, it computes:

\[ \lambda = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2} \]

Then finds eigenvectors by solving:

\[ (A - \lambda I)\mathbf{v} = 0 \]

3. Mathematical Background

Properties:

The sum of eigenvalues equals the trace of the matrix
The product of eigenvalues equals the determinant
A matrix is diagonalizable if it has n linearly independent eigenvectors
Symmetric matrices have real eigenvalues and orthogonal eigenvectors

4. Using the Calculator

Instructions:

Enter your square matrix in the text area
Separate columns with commas (,)
Separate rows with semicolons (;)
Click "Calculate" to compute eigenvalues and eigenvectors

5. Frequently Asked Questions (FAQ)

Q1: What matrices have eigenvalues?
A: Only square matrices have eigenvalues and eigenvectors.

Q2: Can a matrix have complex eigenvalues?
A: Yes, non-symmetric real matrices can have complex eigenvalues (in conjugate pairs).

Q3: What does it mean if an eigenvalue is zero?
A: A zero eigenvalue indicates the matrix is singular (non-invertible).

Q4: How many eigenvectors can one eigenvalue have?
A: An eigenvalue has at least one eigenvector, but can have more (geometric multiplicity).

Q5: What are eigenvalues used for?
A: Applications include stability analysis, principal component analysis (PCA), quantum mechanics, vibration analysis, and more.