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In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix.
Contents
- Description
- Examples
- Unitriangular matrix
- Strictly triangular matrix
- Atomic triangular matrix
- Special properties
- Triangularisability
- Simultaneous triangularisability
- Generalizations
- Borel subgroups and Borel subalgebras
- Forward and back substitution
- Forward substitution
- Algorithm
- Applications
- Glossary
- References
Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if 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. The variable L (standing for lower or left) is commonly used to represent a lower triangular matrix, while the variable U (standing for upper) or R (standing for right) is commonly used for upper triangular matrix. A matrix that is both upper and lower triangular is diagonal.
Matrices that are similar to triangular matrices are called triangularisable.
Many operations on upper triangular matrices preserve the shape:
Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra 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 of square matrices of a fixed size, where the Lie bracket [a,b] given by the commutator ab-ba. The Lie algebra of all upper triangular matrices is often referred to as a Borel subalgebra 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.
Examples
This matrix
is upper triangular and this matrix
is lower triangular.
Unitriangular matrix
If the entries on the main diagonal of a (upper or lower) triangular matrix are all 1, the matrix is called (upper or lower) unitriangular. All unitriangular matrices are unipotent. Other names used for these matrices are unit (upper or lower) triangular (of which "unitriangular" might be a contraction), or very rarely normed (upper or lower) triangular. However a unit triangular matrix is not the same as the unit matrix, and a normed triangular matrix has nothing to do with the notion of matrix norm. The identity matrix is the only matrix which is both upper and lower unitriangular.
The set of unitriangular matrices forms a Lie group.
Strictly triangular matrix
If the entries on the main diagonal of a (upper or lower) triangular matrix are all 0, the matrix is called strictly (upper or lower) triangular. All strictly triangular matrices are nilpotent, and the set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted
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.
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. So an atomic lower triangular matrix is of the form
The inverse of an atomic triangular matrix is again atomic triangular. Indeed, we have
i.e., the off-diagonal entries are replaced in the inverse matrix by their additive inverses.
Examples
The matrix
is atomic lower triangular. Its inverse is
Special properties
A matrix which is simultaneously triangular and normal is also diagonal. This can be seen by looking at the diagonal entries of A*A and AA*, where A is a normal, triangular matrix.
The transpose of an upper triangular matrix is a lower triangular matrix and vice versa.
The determinant of a triangular matrix equals the product of the diagonal entries. Since for any triangular matrix A the matrix
Triangularisability
A matrix that is similar to a triangular matrix is referred to as triangularizable. Abstractly, this is equivalent to stabilizing a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis
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) 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 stabilises 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. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.
Simultaneous triangularisability
A set of matrices
The basic result is that (over an algebraically closed field), the commuting matrices
The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert's Nullstellensatz: commuting matrices form a commutative algebra
This is generalized by Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.
More generally and precisely, a set of matrices
Generalizations
Because the product of two upper triangular matrices is again upper triangular, the set of upper triangular matrices forms an algebra. Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces.
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.
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, which is a subgroup of the general linear group of all invertible matrices. Note that a triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).
Over the real numbers, this group is disconnected, having
The Lie algebra 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 B of the Lie group GLn and the standard Borel subalgebra
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 subgroups. 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 subgroups.
Examples
The group of 2 by 2 upper unitriangular matrices is isomorphic to the additive group of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic Möbius transformations; the 3 by 3 upper unitriangular matrices form the Heisenberg group.
Forward and back substitution
A matrix equation in the form
Notice that this does not require inverting the matrix.
Forward substitution
The matrix equation Lx = b can be written as a system of linear equations
Observe that the first equation (
The resulting formulas are:
A matrix equation with an upper triangular matrix U can be solved in an analogous way, only working backwards.
Algorithm
The following is an example implementation of this algorithm in the C# programming language. Note that the algorithm performs poorly in C# due to the inefficient handling of non-jagged matrices in this language. Nonetheless, the method of forward and backward substitution can be highly efficient.
Applications
Forward substitution is used in financial bootstrapping to construct a yield curve.