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In the
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
discipline of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, a triangular matrix is a special kind of
square matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. A square matrix is called if all the entries ''above'' the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
are zero. Similarly, a square matrix is called if all the entries ''below'' the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
. By the
LU decomposition In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix (mathematics), matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutat ...
algorithm, an
invertible matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
all its leading principal
minors Minor may refer to: * Minor (law) In law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenc ...
are non-zero.


Description

A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form :U = \begin u_ & u_ & u_ & \ldots & u_ \\ & u_ & u_ & \ldots & u_ \\ & & \ddots & \ddots & \vdots \\ & & & \ddots & u_ \\ 0 & & & & u_ \end is called an upper triangular matrix or right triangular matrix. A lower or left triangular matrix is commonly denoted with the variable ''L'', and an upper or right triangular matrix is commonly denoted with the variable ''U'' or ''R''. A matrix that is both upper and lower triangular is
diagonal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
. Matrices that are similar to triangular matrices are called triangularisable. A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a
trapezoid In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's meth ...

trapezoid
.


Examples

This matrix :\begin 1 & 4 & 1 \\ 0 & 6 & 4 \\ 0 & 0 & 1 \\ \end is upper triangular and this matrix :\begin 1 & 0 & 0 \\ 2 & 96 & 0 \\ 4 & 9 & 69 \\ \end is lower triangular.


Forward and back substitution

A matrix equation in the form L\mathbf = \mathbf or U\mathbf = \mathbf is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes x_1, then substitutes that ''forward'' into the ''next'' equation to solve for x_2, and repeats through to x_n. In an upper triangular matrix, one works ''backwards,'' first computing x_n, then substituting that ''back'' into the ''previous'' equation to solve for x_, and repeating through x_1. Notice that this does not require inverting the matrix.


Forward substitution

The matrix equation ''L''x = b can be written as a system of linear equations :\begin \ell_ x_1 & & & & & & & = & b_1 \\ \ell_ x_1 & + & \ell_ x_2 & & & & & = & b_2 \\ \vdots & & \vdots & & \ddots & & & & \vdots \\ \ell_ x_1 & + & \ell_ x_2 & + & \dotsb & + & \ell_ x_m & = & b_m \\ \end Observe that the first equation (\ell_ x_1 = b_1) only involves x_1, and thus one can solve for x_1 directly. The second equation only involves x_1 and x_2, and thus can be solved once one substitutes in the already solved value for x_1. Continuing in this way, the k-th equation only involves x_1,\dots,x_k, and one can solve for x_k using the previously solved values for x_1,\dots,x_. The resulting formulas are: :\begin x_1 &= \frac, \\ x_2 &= \frac, \\ &\ \ \vdots \\ x_m &= \frac. \end A matrix equation with an upper triangular matrix ''U'' can be solved in an analogous way, only working backwards.


Applications

Forward substitution is used in financial
bootstrapping In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input. Etymology Tall boot A boot, plural boots, is a type of specific footwear Footwear refers to garments wor ...
to construct a
yield curve In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated ...

yield curve
.


Properties

The
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

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 (meaning ''A''*''A'' = ''AA''*, where ''A''* is the
conjugate transpose In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
) and triangular is also diagonal. This can be seen by looking at the diagonal entries of ''A''*''A'' and ''AA''*. The
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
and
permanent Permanent may refer to: Art and entertainment *Permanent (film), ''Permanent'' (film), a 2017 American film *Permanent (Joy Division album), ''Permanent'' (Joy Division album) *Permanent (song), "Permanent" (song), by David Cook Other uses *Perm ...
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 In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

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 In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...
, that is, its multiplicity as a root of the
characteristic polynomial In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
p_A(x)=\det(xI-A) of ''A''. In other words, the characteristic polynomial of a triangular ''n''×''n'' matrix ''A'' is exactly : p_A(x) = (x-a_)(x-a_)\cdots(x-a_), that is, the unique degree ''n'' polynomial whose roots are the diagonal entries of ''A'' (with multiplicities). To see this, observe that xI-A is also triangular and hence its determinant \det(xI-A) is the product of its diagonal entries (x-a_)(x-a_)\cdots(x-a_).


Special forms


Unitriangular matrix

If the entries on the
main diagonal In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
of a (upper or lower) triangular matrix are all 1, the matrix is called (upper or lower) unitriangular. Other names used for these matrices are unit (upper or lower) triangular, or very rarely normed (upper or lower) triangular. However, a ''unit'' triangular matrix is not the same as the ''
unit matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by ''I'n'', or simply by ''I'' if the size is immaterial or can be trivially determined b ...

unit matrix
'', and a ''normed'' triangular matrix has nothing to do with the notion of
matrix norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrix (mathematics), matrices (of given dimensions). Preliminaries Given a field (mathematics), field K of either real number, real or complex number ...
. All finite unitriangular matrices are
unipotent In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


Strictly triangular matrix

If all of the entries on the main diagonal of a (upper or lower) triangular matrix are also 0, the matrix is called strictly (upper or lower) triangular. All finite strictly triangular matrices are
nilpotent In mathematics, an element ''x'' of a ring (mathematics), ring ''R'' is called nilpotent if there exists some positive integer ''n'', called the index (or sometimes the degree), such that ''x'n'' = 0. The term was introduced by Benj ...
of index ''n'' as a consequence of the Cayley-Hamilton theorem.


Atomic triangular matrix

An atomic (upper or lower) triangular matrix is a special form of unitriangular matrix, where all of the
off-diagonal element In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
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.


Triangularisability

A matrix that is similar to a triangular matrix is referred to as triangularizable. Abstractly, this is equivalent to stabilizing a
flag A flag is a piece of fabric A textile is a flexible material made by creating an interlocking network of yarn Yarn is a long continuous length of interlocked fibres, suitable for use in the production of textiles, sewing, crochetin ...
: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis (e_1,\ldots,e_n) and the resulting flag 0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n. All flags are conjugate (as the general linear group acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilizes the standard flag. Any complex square matrix is triangularizable. In fact, a matrix ''A'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
containing all of the eigenvalues of ''A'' (for example, any matrix over an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) 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 In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
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. In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix ''A'' has a
Schur decomposition In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. This means that ''A'' is unitarily equivalent (i.e. similar, using a
unitary matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.


Simultaneous triangularisability

A set of matrices A_1, \ldots, A_k are said to be if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix ''P.'' Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the A_i, denoted K _1,\ldots,A_k Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable Lie algebra, solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of ...
. The basic result is that (over an algebraically closed field), the
commuting matrices In linear algebra, two matrix (mathematics), matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set (mathematics), set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, ...
A,B or more generally A_1,\ldots,A_k are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at
commuting matrices In linear algebra, two matrix (mathematics), matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set (mathematics), set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, ...
. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices. The fact that commuting matrices have a common eigenvector can be interpreted as a result of
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''Satz ' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

Hilbert's Nullstellensatz
: commuting matrices form a commutative algebra K _1,\ldots,A_k/math> over K _1,\ldots,x_k/math> which can be interpreted as a variety in ''k''-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an
algebra representation In abstract algebra, a representation of an associative algebra is a module (mathematics), module for that algebra. Here an associative algebra is a (not necessarily unital algebra, unital) ring (mathematics), ring. If the algebra is not unital, i ...
of the polynomial algebra in ''k'' variables. This is generalized by
Lie's theorem In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a Lie algebra, representation ...
, which shows that any representation of a
solvable Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is simultaneously upper triangularizable, the case of commuting matrices being the
abelian Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
case, abelian being a fortiori solvable. More generally and precisely, a set of matrices A_1,\ldots,A_k is simultaneously triangularisable if and only if the matrix p(A_1,\ldots,A_k) _i,A_j/math> is
nilpotent In mathematics, an element ''x'' of a ring (mathematics), ring ''R'' is called nilpotent if there exists some positive integer ''n'', called the index (or sometimes the degree), such that ''x'n'' = 0. The term was introduced by Benj ...
for all polynomials ''p'' in ''k'' ''non''-commuting variables, where _i,A_j/math> is the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
; for commuting A_i the commutator vanishes so this holds. This was proven in ; a brief proof is given in . One direction is clear: if the matrices are simultaneously triangularisable, then _i, A_j/math> is ''strictly'' upper triangularizable (hence nilpotent), which is preserved by multiplication by any A_k or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.


Algebras of triangular matrices

Upper triangularity is preserved by many operations: * The sum of two upper triangular matrices is upper triangular. * The product of two upper triangular matrices is upper triangular. * The inverse of an upper triangular matrix, if it exists, is upper triangular. * The product of an upper triangular matrix and a scalar is upper triangular. Together these facts mean that the upper triangular matrices form a
subalgebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of the
associative algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of square matrices of a fixed size, where the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \righta ...
'a'', ''b''given by the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. The Lie algebra of all upper triangular matrices is a
solvable Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. It is often referred to as a
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable Lie algebra, solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of ...
of the Lie algebra of all square matrices. All these results hold if ''upper triangular'' is replaced by ''lower triangular'' throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either. The set of unitriangular matrices forms a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. The set of strictly upper (or lower) triangular matrices forms a
nilpotent Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, denoted \mathfrak. This algebra is the derived Lie algebra of \mathfrak, the Lie algebra of all upper triangular matrices; in symbols, \mathfrak = mathfrak,\mathfrak In addition, \mathfrak is the Lie algebra of the Lie group of unitriangular matrices. In fact, by
Engel's theorem In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra if and only if for each X \in \mathfrak g, the adjoint representation of a Lie algebra, adjoint m ...
, any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable. Algebras of upper triangular matrices have a natural generalization in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
which yields
nest algebra In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by and have many interesting properties. They are ...
s on
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s.


Borel subgroups and Borel subalgebras

The set of invertible triangular matrices of a given kind (upper or lower) forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, indeed a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, which is a subgroup of the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 2^n 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 In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product of groups, direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product ...
of this group and the group of
diagonal matrices In linear algebra, a diagonal matrix is a matrix (mathematics), matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example ...
with \pm 1 on the diagonal, corresponding to the components. The
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a
solvable Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. These are, respectively, the standard
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski topology, Zariski closed and connected solvable group, solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' inv ...
''B'' of the Lie group GL''n'' and the standard
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable Lie algebra, solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of ...
\mathfrak of the Lie algebra gl''n''. The upper triangular matrices are precisely those that stabilize the standard flag. The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski topology, Zariski closed and connected solvable group, solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' inv ...
s. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order. The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are ''not'' all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called
parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski topology, Zariski closed and connected solvable group, solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' inv ...
s.


Examples

The group of 2×2 upper unitriangular matrices is
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphic
to the
additive group An additive group is a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be ...
of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers sa ...
s; the 3×3 upper unitriangular matrices form the
Heisenberg group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
.


See also

*
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Al ...
*
QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a matrix decomposition, decomposition of a matrix (mathematics), matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and ...
*
Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a matrix decomposition, decomposition of a Hermitian matrix, Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugat ...
*
Hessenberg matrix In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" Triangular matrix, triangular. To be exact, an upper Hessenberg matrix has zero entries below the first diagonal#Matrices, subdiagonal, and a lower Hess ...
*
Tridiagonal matrix In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal only. For example, the following matrix is tridiagonal: :\begin ...
*
Invariant subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...


References

* * * * {{Matrix classes Numerical linear algebra Matrices