In the

^{*}''A'' = ''AA''^{*}, where ''A''^{*} is the ^{*}''A'' and ''AA''^{*}.
The

_{''n''} and the standard _{''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

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 :$L\; =\; \backslash begin\; \backslash ell\_\; \&\; \&\; \&\; \&\; 0\; \backslash \backslash \; \backslash ell\_\; \&\; \backslash ell\_\; \&\; \&\; \&\; \backslash \backslash \; \backslash ell\_\; \&\; \backslash ell\_\; \&\; \backslash ddots\; \&\; \&\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash ddots\; \&\; \backslash \backslash \; \backslash ell\_\; \&\; \backslash ell\_\; \&\; \backslash ldots\; \&\; \backslash ell\_\; \&\; \backslash ell\_\; \backslash end$ is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form :$U\; =\; \backslash begin\; u\_\; \&\; u\_\; \&\; u\_\; \&\; \backslash ldots\; \&\; u\_\; \backslash \backslash \; \&\; u\_\; \&\; u\_\; \&\; \backslash ldots\; \&\; u\_\; \backslash \backslash \; \&\; \&\; \backslash ddots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; \&\; \&\; \&\; \backslash ddots\; \&\; u\_\; \backslash \backslash \; 0\; \&\; \&\; \&\; \&\; u\_\; \backslash end$ 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 isdiagonal
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 ...

.
Examples

This matrix :$\backslash begin\; 1\; \&\; 4\; \&\; 1\; \backslash \backslash \; 0\; \&\; 6\; \&\; 4\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \backslash \backslash \; \backslash end$ is upper triangular and this matrix :$\backslash begin\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 2\; \&\; 96\; \&\; 0\; \backslash \backslash \; 4\; \&\; 9\; \&\; 69\; \backslash \backslash \; \backslash end$ is lower triangular.Forward and back substitution

A matrix equation in the form $L\backslash mathbf\; =\; \backslash mathbf$ or $U\backslash mathbf\; =\; \backslash mathbf$ 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 :$\backslash begin\; \backslash ell\_\; x\_1\; \&\; \&\; \&\; \&\; \&\; \&\; \&\; =\; \&\; b\_1\; \backslash \backslash \; \backslash ell\_\; x\_1\; \&\; +\; \&\; \backslash ell\_\; x\_2\; \&\; \&\; \&\; \&\; \&\; =\; \&\; b\_2\; \backslash \backslash \; \backslash vdots\; \&\; \&\; \backslash vdots\; \&\; \&\; \backslash ddots\; \&\; \&\; \&\; \&\; \backslash vdots\; \backslash \backslash \; \backslash ell\_\; x\_1\; \&\; +\; \&\; \backslash ell\_\; x\_2\; \&\; +\; \&\; \backslash dotsb\; \&\; +\; \&\; \backslash ell\_\; x\_m\; \&\; =\; \&\; b\_m\; \backslash \backslash \; \backslash end$ Observe that the first equation ($\backslash ell\_\; x\_1\; =\; b\_1$) 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 $x\_1,\backslash dots,x\_k$, and one can solve for $x\_k$ using the previously solved values for $x\_1,\backslash dots,x\_$. The resulting formulas are: :$\backslash begin\; x\_1\; \&=\; \backslash frac,\; \backslash \backslash \; x\_2\; \&=\; \backslash frac,\; \backslash \backslash \; \&\backslash \; \backslash \; \backslash vdots\; \backslash \backslash \; x\_m\; \&=\; \backslash frac.\; \backslash end$ 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 financialbootstrapping
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 ...

.
Properties

Thetranspose
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 ...

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''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''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 ...

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 ...

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 ...

$p\_A(x)=\backslash det(xI-A)$ of ''A''. In other words, the characteristic polynomial of a triangular ''n''×''n'' matrix ''A'' is exactly
: $p\_A(x)\; =\; (x-a\_)(x-a\_)\backslash cdots(x-a\_)$,
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 $\backslash det(xI-A)$ is the product of its diagonal entries $(x-a\_)(x-a\_)\backslash cdots(x-a\_)$.
Special forms

Unitriangular matrix

If the entries on themain 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 ...

'', 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 arenilpotent
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 theoff-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 aflag
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 $(e\_1,\backslash ldots,e\_n)$ and the resulting flag $0\; <\; \backslash left\backslash langle\; e\_1\backslash right\backslash rangle\; <\; \backslash left\backslash langle\; e\_1,e\_2\backslash right\backslash rangle\; <\; \backslash cdots\; <\; \backslash left\backslash langle\; e\_1,\backslash ldots,e\_n\; \backslash right\backslash rangle\; =\; K^n.$ 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 $A\_1,\; \backslash ldots,\; A\_k$ 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 $K;\; href="/html/ALL/s/\_1,\backslash ldots,A\_k.html"\; ;"title="\_1,\backslash ldots,A\_k">\_1,\backslash ldots,A\_k$ 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 aBorel 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 $A\_1,\backslash ldots,A\_k$ 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 ...

: commuting matrices form a commutative algebra $K;\; href="/html/ALL/s/\_1,\backslash ldots,A\_k.html"\; ;"title="\_1,\backslash ldots,A\_k">\_1,\backslash ldots,A\_k$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 $A\_1,\backslash ldots,A\_k$ is simultaneously triangularisable if and only if the matrix $p(A\_1,\backslash ldots,A\_k);\; href="/html/ALL/s/\_i,A\_j.html"\; ;"title="\_i,A\_j">\_i,A\_j$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 $;\; href="/html/ALL/s/\_i,A\_j.html"\; ;"title="\_i,A\_j">\_i,A\_j$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 $;\; href="/html/ALL/s/\_i,\_A\_j.html"\; ;"title="\_i,\; A\_j">\_i,\; A\_j$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 asubalgebra 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 $\backslash mathfrak.$ This algebra is the derived Lie algebra of $\backslash mathfrak$, the Lie algebra of all upper triangular matrices; in symbols, $\backslash mathfrak\; =;\; href="/html/ALL/s/mathfrak,\backslash mathfrak.html"\; ;"title="mathfrak,\backslash mathfrak">mathfrak,\backslash mathfrak$ In addition, $\backslash mathfrak$ 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 agroup
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 $\backslash pm\; 1$ 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 GLBorel 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 ...

$\backslash mathfrak$ of the Lie algebra glBorel 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 isisomorphic
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 ...

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