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TL;DR

Non-zero vectors \( \mathbf{v} \) satisfying \( A\mathbf{v} = \lambda\mathbf{v} \). They define directions that are only scaled (not rotated) by the linear transformation A.

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

Eigenvectors

Non-zero vectors \( \mathbf{v} \) satisfying \( A\mathbf{v} = \lambda\mathbf{v} \). They define directions that are only scaled (not rotated) by the linear transformation A.

Why it matters for interviews

Eigenvectors of the covariance matrix are the principal components -- the uncorrelated risk factors. Understanding eigenvectors is essential for PCA-based factor models and portfolio construction.

Definition and Mathematical Foundation

Non-zero vectors \( \mathbf{v} \) satisfying \( A\mathbf{v} = \lambda\mathbf{v} \). They define directions that are only scaled (not rotated) by the linear transformation A.

Application in Quantitative Finance

Eigenvectors of the covariance matrix are the principal components -- the uncorrelated risk factors. Understanding eigenvectors is essential for PCA-based factor models and portfolio construction.

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Frequently Asked Questions

How do eigenvectors define principal components?
The first principal component is the eigenvector with the largest eigenvalue -- the direction of maximum variance. Subsequent components are orthogonal directions of decreasing variance.
Are eigenvectors unique?
Eigenvectors are unique up to scalar multiplication. For distinct eigenvalues, eigenvectors are linearly independent. For repeated eigenvalues, there is freedom in choosing eigenvectors within the eigenspace.
What is the geometric interpretation of eigenvectors?
An eigenvector defines an invariant direction under the transformation A. The corresponding eigenvalue is the scaling factor. For covariance matrices, eigenvectors point along the axes of the data ellipsoid.