Triangular matrix: Difference between revisions

Revision as of 22:53, 13 April 2014

Revision as of 07:51, 14 April 2014

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* The product of an upper triangular matrix by a constant is an upper triangular matrix.

Together these facts mean that the upper triangular matrices form a [[subalgebra]] of the [[associative algebra]] of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the [[Lie algebra]] of square matrices of a fixed size, where the [[Lie bracket]] [''a'',''b''] given by the [[Commutator#Ring_theory|commutator]] ''ab-ba''. The Lie algebra of all upper triangular matrices is often referred to as thea [[Borel subalgebra]] and is denoted <math>\mathfrak{b}</math>. All these results hold if "upper triangular" is replaced by "lower triangular" throughout, soof the lower triangular matrices also form a Lie subalgebra. However, they may not be mixed together (in the first two rules with two operands); for example, the sumalgebra of anall uppersquare and a lower triangular matrix can be any matrixmatrices.

All these results hold if "upper triangular" is replaced by "lower triangular" throughout; in particular the lower triangular matrices also form a Lie algebra. However, noteoperations thatmixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a ''lower'' triangular with an ''upper'' triangular matrix is not necessarily triangular either.

===Examples===