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Tosh5457
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I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?
micromass said:Wiki knows all: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
Check under section "history"
Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.[11] In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.[12] Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.[13]
Tosh5457 said:I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?
The definition of eigenvalues and eigenvectors is important in linear algebra as they provide a way to understand and analyze the behavior of linear transformations. They also have numerous applications in fields such as physics, engineering, and computer science.
In data analysis, eigenvalues and eigenvectors are used to reduce the dimensionality of data sets and identify important patterns and relationships among variables. This allows for easier interpretation and visualization of complex data.
The terms "eigenvalue" and "eigenvector" were chosen by the mathematician David Hilbert in the early 20th century. "Eigen" is a German word meaning "characteristic" or "inherent," and these terms accurately describe the unique properties of these mathematical concepts.
Eigenvalues and eigenvectors are closely related to linear independence. In fact, eigenvectors are linearly independent from each other and can be used to form a basis for the vector space in which they exist. This makes them useful in solving systems of linear equations.
Yes, eigenvalues and eigenvectors can have complex values. This is especially relevant in quantum mechanics, where complex numbers are used to represent the state of a physical system. Complex eigenvalues and eigenvectors can also provide more information about the behavior of a linear transformation.