Triangular matrix: Difference between revisions

Revision as of 16:39, 6 October 2024

Latest revision as of 04:24, 20 August 2025

(11 intermediate revisions by 6 users not shown)

Line 38:

1 & 0 & 0 \\

2 & 96 & 0 \\

4 & 9 & 69 \\

\end{bmatrix}</math>

is the lower triangular, andfor the non symmetric matrix:

:<math>\begin{bmatrix}

1 & 5 & 8 \\

2 & 96 & 9 \\

4 & 9 & 69

\end{bmatrix}</math>

and

:<math>\begin{bmatrix}

1 & 4 & 1 \\

0 & 6 & 9 \\

0 & 0 & 1 \\

\end{bmatrix}</math>

is the upper triangular. for the non symmetric matrix:

:<math>\begin{bmatrix}

1 & 4 & 1 \\

99 & 6 & 9 \\

40 & 88 & 1

\end{bmatrix}</math>

==Forward and back substitution==

Line 92 ⟶ 106:

: <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>

==Special forms==

=== Unitriangular matrix ===

If the entries on the [[main diagonal]] of a (upperlower or lowerupper) triangular matrix are all 1, the matrix is called (upperlower or lowerupper) '''unitriangular'''.

Other names used for these matrices are '''unit''' (upperlower or lowerupper) '''triangular''', or very rarely '''normed''' (upperlower or lowerupper) '''triangular'''. However, a ''unit'' triangular matrix is not the same as '''the''' ''[[identity matrix|unit matrix]]'', and a ''normed'' triangular matrix has nothing to do with the notion of [[matrix norm]].

All finite unitriangular matrices are [[unipotent]].

Line 105 ⟶ 119:

=== Strictly triangular matrix ===

If all of the entries on the main diagonal of a (upperlower or lowerupper) triangular matrix are also 0, the matrix is called '''strictly''' (upperlower or lowerupper) '''triangular'''.

All finite strictly triangular matrices are [[nilpotent matrix|nilpotent]] of index at most ''n'' as a consequence of the [[Cayley–Hamilton theorem|Cayley-Hamilton theorem]].

Line 111 ⟶ 125:

=== Atomic triangular matrix ===

{{Main|Frobenius matrix}}

An '''atomic''' (upperlower or lowerupper) '''triangular matrix''' is a special form of unitriangular matrix, where all of the [[off-diagonal element]]s are zero, except for the entries in a single column. Such a matrix is also called a '''Frobenius matrix''', a '''Gauss matrix''', or a '''Gauss transformation matrix'''.

=== Block triangular matrix ===

Line 127 ⟶ 141:

\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====

Line 146 ⟶ 160:

Any complex square matrix is triangularizable.<ref name="axler"/> In fact, a matrix ''A'' over a [[field (mathematics)|field]] containing all of the eigenvalues of ''A'' (for example, any matrix over an [[algebraically closed field]]) is similar to a triangular matrix. This can be proven by using induction on the fact that ''A'' has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that ''A'' stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.

A more precise statement is given by the [[Jordan normal form]] theorem, which states that in this situation, ''A'' is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.<ref name="axler"/><ref name="herstein">{{Cite book | last = Herstein | first = I. N. | url = https://www.worldcat.org/oclc/3307396 | title = Topics in Algebra | date = 1975 | publisher = Wiley | isbn = 0-471-01090-1 | edition = 2nd | ___location = New York | oclc = 3307396 | pages = 285&ndash;290}}</ref>

In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix ''A'' has a [[Schur decomposition]]. This means that ''A'' is unitarily equivalent (i.e. similar, using a [[unitary matrix]] as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.

Line 160 ⟶ 174:

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 ==

Line 187 ⟶ 201:

===Borel subgroups and Borel subalgebras===

{{main|Borel subgroup|Borel subalgebra}}

The set of invertible triangular matrices of a given kind (upperlower or lowerupper) forms a [[group (mathematics)|group]], indeed a [[Lie group]], which is a subgroup of the [[general linear group]] of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).

Over the real numbers, this group is disconnected, having <math>2^n</math> components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a [[semidirect product]] of this group and the group of [[Diagonal matrix|diagonal matrices]] with <math>\pm 1</math> on the diagonal, corresponding to the components.

Line 214 ⟶ 228:

[[Category:Numerical linear algebra]]

[[Category:Matrices (mathematics)]]