Triangular matrix: Difference between revisions

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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>

The [[matrix exponential]] of a triangular matrix is equal to the scalar [[exponential]] applied to each of its entries.

==Special forms==