eigenvalue

An eigenvalue is a scalar quantity associated with a square matrix or a linear transformation that describes how certain vectors, called eigenvectors, are scaled during the transformation. Specifically, if \(A\) is a square matrix and \(\mathbf{v}\) is a non-zero vector such that when \(A\) acts on \(\mathbf{v}\), the result is a scalar multiple of \(\mathbf{v}\), then this scalar multiple is the eigenvalue corresponding to \(\mathbf{v}\). Mathematically, this is expressed as \(A\mathbf{v} = \lambda\mathbf{v}\), where \(\lambda\) is the eigenvalue.

Eigenvalues are fundamental in various fields such as physics, engineering, and mathematics because they reveal intrinsic properties of systems described by matrices. They are used in stability analysis, modal analysis, quantum mechanics, and principal component analysis, among others.

The eigenvalues of a matrix can be determined by solving the characteristic equation, which is obtained by subtracting \(\lambda\) times the identity matrix from \(A\) and setting the determinant to zero: \(\det(A - \lambda I) = 0\). The solutions to this polynomial equation are the eigenvalues of the matrix.

Eigenvalues are complex numbers in general, even when dealing with real matrices. Their magnitude and distribution give insights into the behavior of the linear systems, such as stability and oscillation. The associated eigenvectors provide directions in which the transformation acts simply by scaling.

Understanding eigenvalues and eigenvectors is essential for analyzing linear transformations and systems, enabling simplified representations and deeper insights into their structure and behavior.