Matrices - Revision history

Patapom at 09:00, 5 August 2017

2017-08-05T09:00:54Z

2017-07-30T18:40:39Z

2017-07-30T18:40:21Z

2017-07-30T18:05:28Z

2017-07-30T17:57:06Z

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← Older revision		Revision as of 09:00, 5 August 2017
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		+	http://www.seas.ucla.edu/~vandenbe/133A/
		+
		+
	== Matrix Decomposition Methods ==		== Matrix Decomposition Methods ==

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 Computers usually ''solve square systems of linear equations'' using the LU decomposition, and it is also a key step when ''inverting a matrix'', or ''computing the determinant of a matrix''.
-== Cholesky decomposition ==
+=== Cholesky decomposition ===
 [https://en.wikipedia.org/wiki/Cholesky_decomposition Cholesky Decomposition] => TODO

← Older revision		Revision as of 18:40, 30 July 2017
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	Computers usually ''solve square systems of linear equations'' using the LU decomposition, and it is also a key step when ''inverting a matrix'', or ''computing the determinant of a matrix''.		Computers usually ''solve square systems of linear equations'' using the LU decomposition, and it is also a key step when ''inverting a matrix'', or ''computing the determinant of a matrix''.
		+
		+	== Cholesky decomposition ==
		+	[https://en.wikipedia.org/wiki/Cholesky_decomposition Cholesky Decomposition] => TODO
		+
		+	==== Applications ====
		+
		+	Useful for efficient numerical solutions and ''Monte Carlo simulations''. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for ''solving systems of linear equations''.

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 === QR Decomposition ===
-QR Decomposition, or QR Factorization is the process of decomposing a '''square''' matrix A this way:
+[https://en.wikipedia.org/wiki/QR_decomposition QR Decomposition], or QR Factorization is the process of decomposing a '''square''' matrix A this way:
 <math>\left [ \bold{A} \right ] = \left [ \bold{Q} \right ] \cdot \left [ \bold{R} \right ]</math>
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 === SVD Decomposition ===
-Singular Value Decomposition (SVD) is the process of decomposing a matrix A this way:
+[https://en.wikipedia.org/wiki/Singular_value_decomposition Singular Value Decomposition] (SVD) is the process of decomposing a matrix A this way:
 <math>\left [ \bold{A} \right ] = \left [ \bold{U} \right ] \cdot \left [ \bold{\Sigma} \right ] \cdot \left [ \bold{V} \right ]^T</math>
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 === LU Decomposition ===
-LU Decomposition, or LU Factorization can be viewed as the matrix form of Gaussian elimination. It is the process of decomposing a '''square''' matrix A this way:
+[https://en.wikipedia.org/wiki/LU_decomposition LU Decomposition], or LU Factorization can be viewed as the matrix form of Gaussian elimination. It is the process of decomposing a '''square''' matrix A this way:
 <math>\left [ \bold{A} \right ] = \left [ \bold{L} \right ] \cdot \left [ \bold{U} \right ]</math>

@@ Line 49: / Line 49: @@
 === LU Decomposition ===
-LU Decomposition, or LU Factorization is the process of decomposing a '''square''' matrix A this way:
+LU Decomposition, or LU Factorization can be viewed as the matrix form of Gaussian elimination. It is the process of decomposing a '''square''' matrix A this way:
 <math>\left [ \bold{A} \right ] = \left [ \bold{L} \right ] \cdot \left [ \bold{U} \right ]</math>
 Where:

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 Where:
 * '''A''' is an <math>m \times n</math> real or complex matrix
-* '''U''' is an <math>m \times m</math> real or complex unitary matrix
+* '''U''' is an <math>m \times m</math> real or complex [https://en.wikipedia.org/wiki/Unitary_matrix unitary] '''orthonormal''' matrix
 * <math>\bold{\Sigma}</math> is an <math>m \times n</math> rectangular diagonal matrix with non-negative real numbers on the diagonal
-* '''V''' is an <math>n \times n</math> real or complex unitary matrix.
+* '''V''' is an <math>n \times n</math> real or complex unitary '''orthonormal''' matrix.
-The diagonal entries <math>\sigma_i</math> of <math>\bold{\Sigma}</math> are known as the singular values of '''A'''. The columns of '''U''' and the columns of '''V''' are called the left-singular vectors and right-singular vectors of '''A''', respectively.
+'''NOTES:'''
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 === LU Decomposition ===

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 The singular value decomposition can be computed using the following observations:
-* The left-singular vectors of M are a set of orthonormal eigenvectors of <math>\bold{A}\bold{A}^T</math>.
+* The left-singular vectors of '''A''' are a set of orthonormal eigenvectors of <math>\bold{A}\bold{A}^T</math>.
-* The right-singular vectors of M are a set of orthonormal eigenvectors of <math>\bold{A}^T\bold{A}</math>.
+* The right-singular vectors of '''A''' are a set of orthonormal eigenvectors of <math>\bold{A}^T\bold{A}</math>.
-* The non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both <math>\bold{A}^T\bold{A}</math> and <math>\bold{A}\bold{A}^T</math>.
+* The non-zero singular values of '''A''' (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both <math>\bold{A}^T\bold{A}</math> and <math>\bold{A}\bold{A}^T</math>.