Triangular matrix: Difference between revisions

Revision as of 00:02, 12 November 2024

Revision as of 10:34, 12 November 2024

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\end{bmatrix}</math>,

where <math>A_{ij} \in \mathbb{F}^{n_i \times n_j}</math> for all <math>i, j = 1, \ldots, k</math>.<ref name="bernstein2009">{{Cite book |last=Bernstein |first=Dennis S. |title=Matrix mathematics: theory, facts, and formulas |date= |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14039-1 |edition=2 |___location=Princeton, NJ |pages=168 |language=en}}</ref>

====Lower block triangular====

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This is generalized by [[Lie's theorem]], which shows that any representation of a [[solvable Lie algebra]] is simultaneously upper triangularizable, the case of commuting matrices being the [[abelian Lie algebra]] case, abelian being a fortiori solvable.

More generally and precisely, a set of matrices <math>A_1,\ldots,A_k</math> is simultaneously triangularisable if and only if the matrix <math>p(A_1,\ldots,A_k)[A_i,A_j]</math> is [[nilpotent]] for all polynomials ''p'' in ''k'' ''non''-commuting variables, where <math>[A_i,A_j]</math> is the [[commutator]]; for commuting <math>A_i</math> the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951;<ref>{{Cite journal | last1 = Drazin | first1 = M. P. | last2 = Dungey | first2 = J. W. | last3 = Gruenberg | first3 = K. W. | date = 1951 | title = Some Theorems on Commutative Matrices |url = http://jlms.oxfordjournals.org/cgi/pdf_extract/s1-26/3/221 | journal = Journal of the London Mathematical Society | language = en | volume = 26 | issue = 3 | pages = 221–228 | doi = 10.1112/jlms/s1-26.3.221}}</ref> a brief proof is given by Prasolov in 1994.<ref>{{Cite book | last = Prasolov | first = V. V. | url = https://www.worldcat.org/oclc/30076024 | title = Problems and Theorems in Linear Algebra | pagepages = 178–179 | date = 1994 | publisher = American Mathematical Society | others = Simeon Ivanov | isbn = 9780821802366 |___location=Providence, R.I. | oclc = 30076024}}</ref> One direction is clear: if the matrices are simultaneously triangularisable, then <math>[A_i, A_j]</math> is ''strictly'' upper triangularizable (hence nilpotent), which is preserved by multiplication by any <math>A_k</math> or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.

== Algebras of triangular matrices ==