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In mathematics, a triangular matrix is a special kind of
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
. A square matrix is called if all the entries ''above'' the
main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
are zero. Similarly, a square matrix is called if all the entries ''below'' the
main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
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 manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
. By the
LU decomposition In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a p ...
algorithm, an
invertible matrix In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
all its leading principal minors 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, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
. 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 A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
.


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 to construct a yield curve.


Properties

The
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
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, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
) and triangular is also diagonal. This can be seen by looking at the diagonal entries of ''A''*''A'' and ''AA''*. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
and permanent 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 transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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, that is, its multiplicity as a root of the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
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, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
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\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
'', and a ''normed'' triangular matrix has nothing to do with the notion of matrix norm. All finite unitriangular matrices are
unipotent In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipoten ...
.


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 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 Benjamin Peirce in the context of his work on the cl ...
of index at most ''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 elements 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 (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
: upper triangular matrices are precisely those that preserve the
standard flag In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space ''V''. Here "increasing" means each is a proper subspace of the next (see filtration): :\ = V_0 \sub V_1 \sub V_ ...
, 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 containing all of the eigenvalues of ''A'' (for example, any matrix over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
) 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. 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 discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper t ...
. This means that ''A'' is unitarily equivalent (i.e. similar, using a
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
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 subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie grou ...
. The basic result is that (over an algebraically closed field), the commuting matrices 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. 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 In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
: 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 for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint func ...
of the polynomial algebra in ''k'' variables. 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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 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 Benjamin Peirce in the context of his work on the cl ...
for all polynomials ''p'' in ''k'' ''non''-commuting variables, where _i,A_j/math> is the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
; for commuting A_i the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951; a brief proof is given by Prasolov in 1994. 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 of the
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
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, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of square matrices of a fixed size, where the Lie bracket 'a'', ''b''given by the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
. The Lie algebra of all upper triangular matrices is a solvable Lie algebra. 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 subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie grou ...
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, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
. The set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, 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, 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, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
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 a ...
s on
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.


Borel subgroups and Borel subalgebras

The set of invertible triangular matrices of a given kind (upper or lower) forms a group, indeed a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, which is a subgroup of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
of this group and the group of diagonal matrices with \pm 1 on the diagonal, corresponding to the components. The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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. 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 closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
''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 subalgebra. The notion is named after Armand Borel. If the Lie algebra \mathfrak is the Lie algebra of a complex Lie grou ...
\mathfrak of the Lie algebra gl''n''. The upper triangular matrices are precisely those that stabilize the
standard flag In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space ''V''. Here "increasing" means each is a proper subspace of the next (see filtration): :\ = V_0 \sub V_1 \sub V_ ...
. 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 closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
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 closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgro ...
s.


Examples

The group of 2×2 upper unitriangular matrices is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the additive group 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 variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
s; the 3×3 upper unitriangular matrices form the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
.


See also

*
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
*
QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix ''A'' into a product ''A'' = ''QR'' of an orthogonal matrix ''Q'' and an upper triangular matrix ''R''. QR decomp ...
* Cholesky decomposition * Hessenberg matrix *
Tridiagonal matrix In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main ...
*
Invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General desc ...


References

{{Matrix classes Numerical linear algebra Matrices