Eigenvectors and Eigenvalues

Definition and Concepts

Eigenvectors are special vectors associated with a matrix, which, when multiplied by a matrix, yield a scaled version of the original vector.
The scalar factor is known as the eigenvalue corresponding to that eigenvector.
More formally, if a matrix A and a vector v satisfy the equation Av = λv, then v is an eigenvector of A and λ is the corresponding eigenvalue.

Calculating Eigenvectors and Eigenvalues

The first step is to determine the eigenvalues of a matrix. This is done by solving the characteristic equation, ** A - λI = 0**, where ‘I’ is an identity matrix.
After the eigenvalues have been found, eigenvectors can be calculated by solving the system of linear equations formed by (A - λI)v = 0.
It’s important to remember that eigenvectors are only determined up to an arbitrary scalar. If v is an eigenvector of A, then so is any scalar multiple of v.

Properties of Eigenvectors and Eigenvalues

Eigenvectors corresponding to distinct eigenvalues are linearly independent.
The determinant of a matrix is equal to the product of its eigenvalues.
The trace (the sum of the entries on the main diagonal) of a matrix is equal to the sum of its eigenvalues.
Eigenvectors and their respective eigenvalues provide insightful information about the matrix, including its determinant, trace and linear transformations it represents.

Applications of Eigenvectors and Eigenvalues

In differential equations, eigenvalues and eigenvectors are used to find solutions to systems of differential equations.
They are also used in the diagonalisation of matrices, a process that simplifies many matrix operations.
In physics, they are used in problems dealing with normal modes of oscillation.
In data analysis, eigenvalues and eigenvectors play an essential role in Principal Component Analysis (PCA), a popular technique for dimensionality reduction.
In network theory, Google’s search algorithm (PageRank) uses concepts of eigenvalues and eigenvectors.

Complex Eigenvalues and Eigenvectors

Eigenvectors and eigenvalues can be complex numbers.
When a matrix has complex eigenvalues, it corresponds to a rotational (as opposed to stretching/squishing) linear transformation.
Complex eigenvalues and eigenvectors follow the same rules as their real-number counterparts.