What is a Eigenvalues Calculator?

An eigenvalues calculator is a tool that computes the eigenvalues of a square matrix. Eigenvalues are scalars that provide insights into the properties of a matrix, particularly in linear transformations and systems of equations.

Key Features:

• Input: Users can enter the elements of the matrix.
• Output: The calculator returns the eigenvalues, which can be real or complex numbers.
• Matrix Size: Typically supports matrices of various sizes (2x2, 3x3, etc.).

Applications:

Eigenvalues are used in various fields, including physics, engineering, and data analysis, particularly in principal component analysis (PCA) and stability analysis.

Matrix:

Using an eigenvalues calculator:

• Input the matrix elements.

Calculation: To find the eigenvalues,
solve the characteristic equation:

det(A – λI) = 0

This results in:

Calculating the determinant:

(4 – λ)(3 – λ) – (2 × 1) = 0
λ² – 7λ + 10 = 0

Factoring gives:

(λ – 5)(λ – 2) = 0

Eigenvalues:

• λ₁ = 5
• λ₂ = 2

So, the eigenvalues of the matrix are 5 and 2.