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Special kind of square matrix

the rings

In mathematics, 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. 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.

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 is used in financial bootstrapping to construct a yield curve.

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. 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, and a normed triangular matrix has nothing to do with the notion of matrix norm. All finite unitriangular matrices are unipotent.

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 of index at most n as a consequence of the Cayley Hamilton theorem.

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.

A block triangular matrix is a block matrix (partitioned matrix) that is a triangular matrix.

A set of matrices are said to be simultaneously triangularisable 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. The basic result is that (over an algebraically closed field), the commuting matrices 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. 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: commuting matrices form a commutative algebra over 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 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 case, abelian being a fortiori solvable. More generally and precisely, a set of matrices is simultaneously triangularisable if and only if the matrix is nilpotent for all polynomials p in k non commuting variables, where is the commutator; for commuting 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 is strictly upper triangularizable (hence nilpotent), which is preserved by multiplication by any or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.

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. 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 semidirect product of this group and the group of diagonal matrices with on the diagonal, corresponding to the components. 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 of the Lie algebra gln. 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.

The group of 2×2 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 3×3 upper unitriangular matrices form the Heisenberg group.