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 matrix. ...
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 matrix. ...
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). It is the study of ...
. 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 pe ...
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 bicondi ...
all its leading principal
minors are non-zero.
Description
A matrix of the form
:
is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form
:
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 polygon, convex quadri ...
.
Examples
This matrix
:
is upper triangular and this matrix
:
is lower triangular.
Forward and back substitution
A matrix equation in the form
or
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
, then substitutes that ''forward'' into the ''next'' equation to solve for
, and repeats through to
. In an upper triangular matrix, one works ''backwards,'' first computing
, then substituting that ''back'' into the ''previous'' equation to solve for
, and repeating through
.
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
:
Observe that the first equation (
) only involves
, and thus one can solve for
directly. The second equation only involves
and
, and thus can be solved once one substitutes in the already solved value for
. Continuing in this way, the
-th equation only involves
, and one can solve for
using the previously solved values for
. The resulting formulas are:
:
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 boots may have a tab, loop or handle at the top known as a bootstrap, allowing one to use fingers ...
to construct a
yield curve
In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...
.
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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
(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 con ...
) 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 and ...
and
permanent
Permanent may refer to:
Art and entertainment
* ''Permanent'' (film), a 2017 American film
* ''Permanent'' (Joy Division album)
* "Permanent" (song), by David Cook
Other uses
* Permanent (mathematics), a concept in linear algebra
* Permanent (cy ...
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 denoted b ...
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 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 denoted b ...
, 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 chara ...
of ''A''. In other words, the characteristic polynomial of a triangular ''n''×''n'' matrix ''A'' is exactly
:
,
that is, the unique degree ''n'' polynomial whose roots are the diagonal entries of ''A'' (with multiplicities).
To see this, observe that
is also triangular and hence its determinant
is the product of its diagonal entries
.
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 matrix. ...
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
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows ...
.
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 unipotent ...
.
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 class ...
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 element
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 δΠ...
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 (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_2 ...
, which is given by the standard ordered basis
and the resulting flag
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 grass ...
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
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to som ...
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 tri ...
. 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
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
denoted
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 group, ...
.
The basic result is that (over an algebraically closed field), the
commuting matrices In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the s ...
or more generally
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 matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the s ...
. 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 ge ...
: commuting matrices form a commutative algebra
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
The set of invertible triangular matrices of a given kind (upper or lower) forms a
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
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
of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a
. These are, respectively, the standard
. 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
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
s.
of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic
.