Prab: CSV import

2025-02-18T11:46:36Z

CSV import

← Older revision		Revision as of 11:46, 18 February 2025
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			<gallery>
			File:Mona_Lisa_eigenvector_grid.png\|Eigenvalues and eigenvectors
			File:Eigenvectors_of_a_linear_operator.gif\|Eigenvectors of a linear operator
			File:Eigenvalue_equation.svg\|Eigenvalue equation
			File:Eigenvectors.gif\|Eigenvectors
			File:Eigenvectors-extended.gif\|Eigenvectors extended
			File:Homothety_in_two_dim.svg\|Homothety in two dimensions
			File:Unequal_scaling.svg\|Unequal scaling
			File:Rotation.png\|Rotation
			File:Shear.svg\|Shear
			File:Squeeze_r=1.5.svg\|Squeeze with r=1.5
			File:GaussianScatterPCA.png\|Gaussian Scatter PCA
			File:Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif\|Mode Shape of a Tuning Fork at Eigenfrequency 440.09 Hz
			</gallery>

Prab: CSV import

2024-03-19T05:52:50Z

CSV import

New page

'''Eigenvalues and eigenvectors''' are fundamental concepts in [[linear algebra]] that play a pivotal role in various areas of mathematics and its applications, including [[differential equations]], [[quantum mechanics]], [[systems theory]], and [[statistics]]. They are particularly important in the analysis and solution of linear systems of equations.

==Definition==
Given a square matrix ''A'', an '''eigenvector''' of ''A'' is a nonzero vector ''v'' such that multiplication by ''A'' alters only the scale of ''v'':
\[A\mathbf{v} = \lambda\mathbf{v}\]
Here, \(\lambda\) is a scalar known as the '''eigenvalue''' associated with the eigenvector ''v''.

==Properties==
* '''Characteristic Polynomial''': The eigenvalues of a matrix ''A'' are the roots of the characteristic polynomial, which is defined as \(\det(A - \lambda I) = 0\), where ''I'' is the identity matrix of the same dimension as ''A''.
* '''Multiplicity''': An eigenvalue's multiplicity is the number of times it is a root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with it.
* '''Diagonalization''': A matrix ''A'' is diagonalizable if there exists a diagonal matrix ''D'' and an invertible matrix ''P'' such that \(A = PDP^{-1}\). The diagonal entries of ''D'' are the eigenvalues of ''A'', and the columns of ''P'' are the corresponding eigenvectors.
* '''Spectral Theorem''': For symmetric matrices, the spectral theorem states that the matrix can be diagonalized by an orthogonal matrix, and its eigenvalues are real.

==Applications==
* In [[quantum mechanics]], eigenvalues and eigenvectors are used to solve the Schrödinger equation, where the eigenvalues represent the energy levels of a quantum system.
* In [[vibration analysis]], the eigenvalues determine the natural frequencies at which structures will resonate.
* Eigenvalues and eigenvectors are used in [[Principal Component Analysis (PCA)]] in statistics for dimensionality reduction and data analysis.
* In [[graph theory]], the eigenvalues of the adjacency matrix of a graph are related to many properties of the graph, such as its connectivity and its number of walks.

==Calculation==
The calculation of eigenvalues and eigenvectors is a fundamental problem in numerical linear algebra. Various algorithms exist for their computation, especially for large matrices, including the QR algorithm, power iteration, and the Jacobi method for symmetric matrices.

==See Also==
* [[Linear algebra]]
* [[Matrix (mathematics)|Matrix theory]]
* [[Spectral theory]]
* [[Numerical linear algebra]]

[[Category:Linear algebra]]
[[Category:Mathematical concepts]]

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← Older revision		Revision as of 11:46, 18 February 2025
Line 31:		Line 31:

	{{math-stub}}		{{math-stub}}
			<gallery>
			File:Mona_Lisa_eigenvector_grid.png\|Eigenvalues and eigenvectors
			File:Eigenvectors_of_a_linear_operator.gif\|Eigenvectors of a linear operator
			File:Eigenvalue_equation.svg\|Eigenvalue equation
			File:Eigenvectors.gif\|Eigenvectors
			File:Eigenvectors-extended.gif\|Eigenvectors extended
			File:Homothety_in_two_dim.svg\|Homothety in two dimensions
			File:Unequal_scaling.svg\|Unequal scaling
			File:Rotation.png\|Rotation
			File:Shear.svg\|Shear
			File:Squeeze_r=1.5.svg\|Squeeze with r=1.5
			File:GaussianScatterPCA.png\|Gaussian Scatter PCA
			File:Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif\|Mode Shape of a Tuning Fork at Eigenfrequency 440.09 Hz
			</gallery>

Eigenvalues and eigenvectors - Revision history

Prab: CSV import

Prab: CSV import