Eigenvalues
- ]University of Sydney, Australia]
- Overview
A scalar value is a value that has only one component, which is its magnitude. For example, speed, height, and mass are all scalar values because they only have one component: how fast you are going, how tall you are, and your mass, respectively. Other examples of scalar values include volume, density, energy, and time.
Eigenvalues are scalar values that are associated with eigenvectors in linear transformations. They are also known as characteristic roots, characteristic values, proper values, or latent roots.
In a linear transformation, an eigenvector is a nonzero vector that does not change direction when the transformation is applied to it. The eigenvalue is the scalar value that scales the eigenvector. The equation for this is called the eigenvalue equation, or eigenequation.
The word "eigen" is German for "characteristic". Eigenvalues are important in many fields, including physics, engineering, computer graphics, and control systems. Some applications include:
- Stability analysis
- The physics of rotating bodies
- Small oscillations of vibrating systems
- Atomic orbitals
- Facial recognition
- Matrix diagonalization
In statistics, eigenvalues are also used in factor analysis to represent the amount of variance contained by a factor. In quantum mechanics, eigenvalues are used to designate the value of a measurable quantity associated with a wavefunction.
[More to come ...]
