Triangular Matrices

If in a matrix \(\,\boldsymbol{L}=[l_{ij}]_{n\times n}\in M_n(K)\,\) all elements located above the main diagonal do vanish: \(\ l_{ij}=0\ \) for \(\ i<j,\ \) \(\ i,j=1,2,\dots,n\,,\ \) then \(\,\boldsymbol{L}\,\) is named \(\,\) a \(\,\) lower triangular \(\,\) matrix.

Correspondingly, \(\,\) in \(\,\) an \(\,\) upper triangular \(\,\) matrix \(\,\boldsymbol{U}=[u_{ij}]_{n\times n}\in M_n(K)\,\) vanishing are all elements below the main diagonal: \(\,u_{ij}=0\ \ \text{for}\ \ i>j,\ \ i,j=1,2,\dots,n\,.\)

For instance, triangular matrices of size \(\,4\,\) have the following general form:

(1)\[\begin{split}\boldsymbol{L}\ =\ \left[\begin{array}{cccc} l_{11} & 0 & 0 & 0 \\ l_{21} & l_{22} & 0 & 0 \\ l_{31} & l_{32} & l_{33} & 0 \\ l_{41} & l_{42} & l_{43} & l_{44} \end{array}\right]\,, \qquad \boldsymbol{U}\ =\ \left[\begin{array}{cccc} u_{11} & u_{12} & u_{13} & u_{14} \\ 0 & u_{22} & u_{23} & u_{24} \\ 0 & 0 & u_{33} & u_{34} \\ 0 & 0 & 0 & u_{44} \end{array}\right]\,.\end{split}\]

Properties of triangular matrices.

• The sum of two lower triangular matrices is a lower triangular matrix.

• The product of a lower triangular matrix by a scalar is a lower triangular matrix.

• The product of two lower triangular matrices is a lower triangular matrix.

• The inverse of a non-singular lower triangular matrix is also lower triangular.

The set of all lower triangular matrices of size \(\,n\,\) over a field \(\,K\,\) is therefore a subalgebra of the algebra \(\,M_n(K).\ \) The same holds for upper triangular matrices.

A system of linear equations with a square coefficient matrix \(\boldsymbol{A}\,\) is easy to solve, when \(\,\boldsymbol{A}\,\) is a lower or upper triangular matrix. For example, let’s consider a system with the matrix \(\,\boldsymbol{L}\,\) in Eq. (1), assuming that \(\ l_{ii}\neq 0,\ \ i=1,2,3,4\,:\)

\begin{alignat*}{5} l_{11}\,x_1 & {\,} {\,} & & {\,} {\,} & & {\,} {\,} & & {\ \ } = {\ \ } & b_1 \\ l_{21}\,x_1 & {\,} + {\,} & l_{22}\,x_2 & {\,} {\,} & & {\,} {\,} & & {\ \ } = {\ \ } & b_2 \\ l_{31}\,x_1 & {\,} + {\,} & l_{32}\,x_2 & {\,} + {\,} & l_{33}\,x_3 & {\,} {\,} & & {\ \ } = {\ \ } & b_3 \\ l_{41}\,x_1 & {\,} + {\,} & l_{42}\,x_2 & {\,} + {\,} & l_{43}\,x_3 & {\,} + {\,} & l_{44}\,x_4 & {\ \ } = {\ \ } & b_4 \end{alignat*}

The solution is readily obtained \(\,\) by \(\,\) forward substitution:

\begin{eqnarray*} x_1 & = & l_{11}^{-1}\ b_1 \\ x_2 & = & l_{22}^{-1}\ (b_2-l_{21}\,x_1) \\ x_3 & = & l_{33}^{-1}\ (b_3-l_{31}\,x_1-l_{32}\,x_2) \\ x_4 & = & l_{44}^{-1}\ (b_4-l_{41}\,x_1-l_{42}\,x_2-l_{43}\,x_3) \end{eqnarray*}

In general, if the coefficient matrix is a lower triangular one of size \(\,n\,\) with non-zero diagonal elements (the latter assures the existence of a unique solution), then

\[x_k\ \,=\ \,l_{kk}^{-1}\ \left(\,b_k\ -\ \sum_{i=1}^{k-1}\ l_{ki}\,x_i\,\right)\,,\qquad k=1,2,\dots,n\,.\]

An analogous procedure, viz. the backward substitution, \(\,\) yields the solution of a linear system with an upper triangular coefficient matrix.