Triangular matrix

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In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve they are very important in numerical analysis. The LU decomposition gives an algorithm to decompose any invertible matrix A into a normed lower triangle matrix L and an upper triangle matrix U.

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[edit] Description

A matrix of the form

\mathbf{L}= \begin{bmatrix} l_{1,1} & & & & 0 \\ l_{2,1} & l_{2,2} & & & \\ l_{3,1} & l_{3,2} & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ l_{n,1} & l_{n,2} & \ldots & l_{n,n-1} & l_{n,n} \end{bmatrix}

is called lower triangular matrix or left triangular matrix, and analogously a matrix of the form

\mathbf{U} = \begin{bmatrix} u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\ & u_{2,2} & u_{2,3} & \ldots & u_{2,n} \\ & & \ddots & \ddots & \vdots \\ & & & \ddots & u_{n-1,n}\\ 0 & & & & u_{n,n} \end{bmatrix}

is called upper triangular matrix or right triangular matrix.

The standard operations on triangular matrices conveniently preserve the triangular form: the sum and product of two upper triangular matrices is again upper triangular. The inverse of an upper triangular matrix is also upper triangular, and of course we can multiply an upper triangular matrix by a constant and it will still be upper triangular. This means that the upper triangular matrices form a subalgebra of the ring of square matrices for any given size. The analogous result holds for lower triangular matrices. Note, however, that the product of a lower triangular with an upper triangular matrix does not preserve triangularity.

[edit] Special forms

[edit] Strictly triangular matrix

A triangular matrix with zero entries on the main diagonal is strictly upper or lower triangular. All strictly triangular matrices are nilpotent.

[edit] Unitriangular matrix

If the entries on the main diagonal are 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 normed upper/lower triangular (very rarely used). 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.

[edit] Gauss matrix

A Gauss matrix is a special form of a unitriangular matrix, where all of the off-diagonal entries are zero, except for the entries in one column. Such a matrix is also called atomic upper/lower triangular or Gauss (transformation) matrix. So an atomic lower triangular matrix is of the form

\mathbf{L}_{i} = \begin{bmatrix} 1 & & & & & & & 0 \\ 0 & \ddots & & & & & & \\ 0 & \ddots & 1 & & & & & \\ 0 & \ddots & 0 & 1 & & & & \\ & & 0 & l_{i+1,i} & 1 & & & \\ \vdots & & 0 & l_{i+2,i} & 0 & \ddots & & \\ & & \vdots & \vdots & \vdots & \ddots & 1 & \\ 0 & \dots & 0 & l_{n,i} & 0 & \dots & 0 & 1 \\ \end{bmatrix}.

The inverse of an atomic triangular matrix is again atomic triangular. Indeed, we have

\mathbf{L}_{i}^{-1} = \begin{bmatrix} 1 & & & & & & & 0 \\ 0 & \ddots & & & & & & \\ 0 & \ddots & 1 & & & & & \\ 0 & \ddots & 0 & 1 & & & & \\ & & 0 & -l_{i+1,i} & 1 & & & \\ \vdots & & 0 & -l_{i+2,i} & 0 & \ddots & & \\ & & \vdots & \vdots & \vdots & \ddots & 1 & \\ 0 & \dots & 0 & -l_{n,i} & 0 & \dots & 0 & 1 \\ \end{bmatrix},

i.e., the off-diagonal entries are replaced by their opposites.

[edit] Special properties

A matrix which is simultaneously upper and lower triangular is diagonal. The identity matrix is the only matrix which is both upper and lower unitriangular.

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 XI − A, whose determinant is the characteristic polynomial of A, is also triangular, the diagonal entries of A in fact give the multiset of eigenvalues of A (an eigenvalue with multiplicity m occurs exactly m times as diagonal entry).[1]

Any complex square matrix is similar to a triangular matrix.[1] 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. 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.[1][2]

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.

The variable L is commonly used for lower triangular matrix, standing for lower/left, while the variable U or R is commonly used for upper triangular matrix, standing for upper/right.

Many operations can be performed more easily on triangular matrices than on general matrices. Notably matrix inversion (when possible) can be done by back substitution, avoiding the complications of general Gaussian elimination.

[edit] 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.

The set of invertible triangular matrices of a given kind (upper or lower) forms a group, which is a subgroup of the general linear group of all invertible matrices. 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.

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 trinagular 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.

[edit] Examples

The matrix

\begin{bmatrix} 1 & 4 & 2 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \\ \end{bmatrix}

is upper triangular and

\begin{bmatrix} 1 & 0 & 0 \\ 2 & 8 & 0 \\ 4 & 9 & 7 \\ \end{bmatrix}

is lower triangular.

The matrix

\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ \end{bmatrix}

is atomic lower triangular. Its inverse is

\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -4 & 1 & 0 \\ 0 & -2 & 0 & 1 \\ \end{bmatrix}.

[edit] Application: back substitution

A matrix equation in the form

\mathbf{L}\mathbf{x} = \mathbf{b}

or

\mathbf{U} \mathbf{x} = \mathbf{b}

is very easy to solve. The matrix equation Lx = b can be written as a system of linear equations

\begin{matrix} l_{1,1} x_1 & & & & & = & b_1 \\ l_{2,1} x_1 & + & l_{2,2} x_2 & & & = & b_2 \\ \vdots & & \vdots & \ddots & & & \vdots \\ l_{m,1} x_1 & + & l_{m,2} x_2 & + \ldots + & l_{m,m} x_m & = & b_m \\ \end{matrix}

which can be solved by the following recursive relation

x_1 = \frac{b_1}{l_{1,1}},
x_2 = \frac{b_2 - l_{2,1} x_1}{l_{2,2}},
\vdots
x_m = \frac{b_m - \sum_{i=1}^{m-1} l_{m,i}x_i}{l_{m,m}}.

A matrix equation with an upper triangular matrix U can be solved in an analogous way. This process is called back substitution.

[edit] See also

[edit] References

  1. ^ a b c Sheldon Axler, Linear Algebra Done Right, Springer-Verlag, 1996, ISBN 0-387-98258-2, pp. 86-87, 169.
  2. ^ I. N. Herstein, Topics in Algebra (2nd edition), John Wiley and Sons, 1975, ISBN 0-471-01090-1, pp. 285-290.