Matrix decomposition: Difference between revisions

Revision as of 21:32, 14 September 2022

Revision as of 21:34, 14 September 2022

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*Related: the [[LDU decomposition|''LDU'' decomposition]] is <math>A=LDU</math>, where ''L'' is [[triangular matrix|lower triangular]] with ones on the diagonal, ''U'' is [[triangular matrix|upper triangular]] with ones on the diagonal, and ''D'' is a [[diagonal matrix]].

*Related: the [[LUP decomposition|''LUP'' decomposition]] is <math>PA=LU</math>, where ''L'' is [[triangular matrix|lower triangular]], ''U'' is [[triangular matrix|upper triangular]], and ''P'' is a [[permutation matrix]].

*Existence: An LUP decomposition exists for any square matrix ''A''. When ''P'' is an [[identity matrix]], the LUP decomposition reduces to the LU decomposition. If the LU decomposition exists, then the LDU decomposition exists.<ref>{{harvnb|Simon|Blume|1994}} Chapter 7.</ref>

*Comments: The LUP and LU decompositions are useful in solving an ''n''-by-''n'' system of linear equations <math>A \mathbf{x} = \mathbf{b}</math>. These decompositions summarize the process of [[Gaussian elimination]] in matrix form. Matrix ''P'' represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the [[row echelon form]] without requiring any row interchanges, then ''P''&nbsp;=&nbsp;''I'', so an LU decomposition exists.