Eigenvalue Calculator (2x2)

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Find the eigenvalues of a 2x2 matrix and understand the trace and determinant.

Matrix Elements

a₁₁

a₁₂

a₂₁

a₂₂

Eigenvalues

3.0000

λ₁

Eigenvalues represent the scaling factors of the linear transformation.

λ₂

1.0000

Trace (Tr)4

Determinant (Det)3

About this calculator

What are Eigenvalues?

An eigenvalue is a scalar associated with a linear system of equations. For a square matrix A, an eigenvalue λ is a value such that there exists a non-zero vector v where Av = λv. This vector v is called the corresponding eigenvector. Eigenvalues reveal fundamental properties of the transformation, such as whether it stretches, shrinks, or rotates space.

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Pro Tips

The sum of eigenvalues equals the Trace of the matrix (a₁₁ + a₂₂).
The product of eigenvalues equals the Determinant (a₁₁a₂₂ - a₁₂a₂₁).
If the discriminant (Tr² - 4*Det) is negative, eigenvalues are complex.

Fun Facts

"Google's PageRank algorithm uses the dominant eigenvalue of a massive link matrix to rank web pages."
"In structural engineering, eigenvalues help determine the natural frequencies of vibration of a building or bridge."
"Quantum mechanics is essentially the study of eigenvalues and eigenvectors of operators on Hilbert spaces."

Formula: det(A - λI) = 0