Triangular 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.
is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices.
In an upper triangular matrix, one works backwards, first computing
Forward substitution is used in financial bootstrapping to construct a yield curve.
In a similar vein, a matrix which is both normal (meaning A*A = AA*, where A* is the conjugate transpose) and triangular is also diagonal.
The determinant 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 eigenvalues of a triangular matrix are exactly its diagonal entries.
All finite strictly triangular matrices are nilpotent of index at most n as a consequence of the Cayley-Hamilton theorem.
An atomic (lower or upper) 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.
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.
The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.
[1][3] 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.
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
The basic result is that (over an algebraically closed field), the commuting matrices
This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before.
As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.
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.
This was proven by Drazin, Dungey, and Gruenberg in 1951;[4] a brief proof is given by Prasolov in 1994.
is strictly upper triangularizable (hence nilpotent), which is preserved by multiplication by any
Upper triangularity is preserved by many operations: Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size.
It is often referred to as a Borel subalgebra of the Lie algebra of all square matrices.
The set of unitriangular matrices forms a Lie group.
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.
The upper triangular matrices are precisely those that stabilize the standard flag.
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.
