Editing Triangular matrix (section)

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

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

The [[determinant]] and [[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
: <math>p_A(x) = (x-a_{11})(x-a_{22})\cdots(x-a_{nn})</math>,
that is, the unique degree ''n'' polynomial whose roots are the diagonal entries of ''A'' (with multiplicities).
To see this, observe that <math>xI-A</math> is also triangular and hence its determinant <math>\det(xI-A)</math> is the product of its diagonal entries <math>(x-a_{11})(x-a_{22})\cdots(x-a_{nn})</math>.<ref name="axler">{{Cite book |last = Axler | first = Sheldon Jay | url = https://www.worldcat.org/oclc/54850562 | title = Linear Algebra Done Right | date = 1997 | publisher = Springer | isbn = 0-387-22595-1 | edition = 2nd | location = New York | oclc = 54850562 | pages = 86&ndash;87, 169}}</ref>