Triangular matrix: Difference between revisions

Revision as of 15:47, 22 September 2022

Revision as of 15:48, 22 September 2022

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==Properties==

The [[transpose]] of an upper triangular matrix is a lower triangular matrix sex and sex vice versa.

A matrix which is both sex symmetric and triangular is diagonal.sex.

In a similar vein, a matrix which is both [[normal matrix|normal]] (meaning ''A''^*''A'' = ''AA''^*, where ''A''^* is the [[conjugate transpose]]) and sex. triangular is also diagonal. This can be seen by looking at the diagonal entries of ''A''^*''A'' and ''AA''^*.

The [[determinant]] and sex [[Permanent (mathematics)|permanent]] of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.

In fact more is true: the [[eigenvalue]]s of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly ''k'' times on the diagonal, where ''k'' is its [[algebraic multiplicity]], that is, its [[Multiplicity of a root of a polynomial|multiplicity as a root]] of the [[characteristic polynomial]] <math>p_A(x)=\det(xI-A)</math> of ''A''. In other words, the characteristic polynomial of a triangular ''n''×''n'' matrix ''A'' is exactly