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TL;DR

Scalars \( \lambda \) satisfying \( A\mathbf{v} = \lambda \mathbf{v} \) for some non-zero vector \( \mathbf{v} \) (the eigenvector). Found by solving \( \det(A - \lambda I) = 0 \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Eigenvalues

Scalars \( \lambda \) satisfying \( A\mathbf{v} = \lambda \mathbf{v} \) for some non-zero vector \( \mathbf{v} \) (the eigenvector). Found by solving \( \det(A - \lambda I) = 0 \).

Why it matters for interviews

Eigenvalues determine the behavior of linear systems, are central to PCA (the primary dimensionality reduction technique in quant finance), and characterize covariance matrix structure for portfolio risk.

Definition and Mathematical Foundation

Scalars \( \lambda \) satisfying \( A\mathbf{v} = \lambda \mathbf{v} \) for some non-zero vector \( \mathbf{v} \) (the eigenvector). Found by solving \( \det(A - \lambda I) = 0 \).

Application in Quantitative Finance

Eigenvalues determine the behavior of linear systems, are central to PCA (the primary dimensionality reduction technique in quant finance), and characterize covariance matrix structure for portfolio risk.

Related Terms

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Frequently Asked Questions

What do eigenvalues of a covariance matrix represent?▼
Each eigenvalue represents the variance explained along the corresponding eigenvector (principal component). Large eigenvalues capture major risk factors; small ones represent noise.
What is the spectral theorem?▼
A real symmetric matrix (like a covariance matrix) has real eigenvalues and orthogonal eigenvectors. It can be decomposed as \( A = Q\Lambda Q^T \) where Q is orthogonal and \( \Lambda \) is diagonal.
How are eigenvalues computed in practice?▼
The QR algorithm iteratively decomposes the matrix until it converges to upper triangular form, revealing eigenvalues on the diagonal. For large sparse matrices, iterative methods like Lanczos are used.