Difference between revisions of "Eigenvectors and Eigenvalues"
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In Physics, eigenvectors and eigenvalues play an important role. For example, solving [[Schrödinger's equation]] (one of the fundamental equations of [[Quantum mechanics]]) is a problem of finding Eigenvectors and Eigenvalues for a very complex linear operator. The ''quantization'' (which means that discrete values are acceptable in physics, but not all of them: for example, an electron can have spin 1/2 or -1/2, but never 1/3 or 1/10) that was previously a postulated of Quantum mechanics can be derived from the fact that the eigenvalues of the operators come in discrete sets. | In Physics, eigenvectors and eigenvalues play an important role. For example, solving [[Schrödinger's equation]] (one of the fundamental equations of [[Quantum mechanics]]) is a problem of finding Eigenvectors and Eigenvalues for a very complex linear operator. The ''quantization'' (which means that discrete values are acceptable in physics, but not all of them: for example, an electron can have spin 1/2 or -1/2, but never 1/3 or 1/10) that was previously a postulated of Quantum mechanics can be derived from the fact that the eigenvalues of the operators come in discrete sets. | ||
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Revision as of 11:55, July 13, 2016
In linear algebra, when a transformation of a space is carried out, some vectors (points in the space) are not rotated, but only extended or shrunk (moved farther or closer to the origin). The vectors are called the eigenvectors of the transformation, and the amount of extension or shrinkage carried out on that eigenvector is called the eigenvalue of the transformation corresponding to the given eigenvector.
Characteristic Property of Eigenvalues and Eigenvectors
An eigenvalue of a square n × n matrix with real entries Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a scalar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \in \mathbb{R}} such that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\boldsymbol{x}=\lambda\boldsymbol{x}}
for some non-zero vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}\in\mathbb{R}^n} known as a eigenvector. The eigenvalues are the zeroes of a matrix's characteristic polynomial, the degree of the corresponding root is called the algebraic multiplicity of the eigenvalue.
The same definition is valid for n × n matrices over any field F: Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \in \mathbf{F}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{x}\in\mathbf{F}^n} .
The eigenvectors represent directions that are preserved by linear transformations of a vector space.
All the eigenvectors of a particular eigenvalue span a vector space called the eigenspace.
If the characteristic polynomial splits into linear factors, then he product of all the eigenvalues of a matrix counted with their algebraic multiplicities equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue.
The span of all the eigenvectors corresponding to a fixed eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is called the eigenspace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\lambda} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . The dimension of this space is called the geometric multiplicity of the eigenvalue.
Eigenvectors and Eigenvalues in Physics
In Physics, eigenvectors and eigenvalues play an important role. For example, solving Schrödinger's equation (one of the fundamental equations of Quantum mechanics) is a problem of finding Eigenvectors and Eigenvalues for a very complex linear operator. The quantization (which means that discrete values are acceptable in physics, but not all of them: for example, an electron can have spin 1/2 or -1/2, but never 1/3 or 1/10) that was previously a postulated of Quantum mechanics can be derived from the fact that the eigenvalues of the operators come in discrete sets.