Eigenvector

TL;DR

• A direction that a matrix maps to a scalar multiple of itself, so the vector’s direction is preserved while its length may change.
• Used to identify principal directions in data (the directions of maximum variance) for compression and dimensionality reduction.
• Appears in operations such as rotations and compressions and is applied across physics, engineering, and computer science.

Definition

An eigenvector is a vector that, when multiplied by a matrix, results in a scalar multiple of the original vector.

Explanation

When a matrix acts on certain vectors, those vectors may remain aligned with their original direction; they are scaled by a constant factor but do not change direction. Such vectors are called eigenvectors of the matrix. This concept is applicable across many fields, including physics, engineering, and computer science, and helps reveal underlying structure and directions of change in a system.

Examples

Axis-aligned vectors under a 2D rotation

One example is a vector whose direction is parallel to one of the axes in a coordinate system. For instance, with a matrix that represents a rotation in two-dimensional space, the eigenvectors would be the vectors that point along the x- and y-axes. When these vectors are multiplied by the matrix, they will still point along the same axes, but their length may be scaled by a constant factor.

Directions of maximum change in data

Another example is a vector that represents the direction of maximum change in a system. In machine learning, eigenvectors are often used to identify the most important features in a dataset. If a matrix represents a set of data points, the eigenvectors will be the vectors that point along the directions of maximum variance in the data. These vectors can then be used to compress the data and reduce the dimensions of the dataset.

Use cases

• Identifying directions of maximum variance in datasets for dimensionality reduction and data compression.
• Analyzing geometric operations such as rotations and compressions.
• Understanding underlying structure and directions of change in physical or engineered systems.

Related terms

• Matrix
• Rotation
• Variance
• Dimensionality reduction
• Machine learning