Editing Triangular matrix (section)

== Description ==
A matrix of the form 
:<math>L = \begin{bmatrix}
  \ell_{1,1} &            &        &              &          0 \\
  \ell_{2,1} & \ell_{2,2} &        &              &            \\
  \ell_{3,1} & \ell_{3,2} & \ddots &              &            \\
      \vdots &     \vdots & \ddots &       \ddots &            \\
  \ell_{n,1} & \ell_{n,2} & \ldots & \ell_{n,n-1} & \ell_{n,n}
\end{bmatrix}</math>

is called a '''lower triangular matrix''' or '''left triangular matrix''', and analogously a matrix of the form
:<math>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}</math>

is called an '''upper triangular matrix''' or '''right triangular matrix'''. A lower or left triangular matrix is commonly denoted with the variable ''L'', and an upper or right triangular matrix is commonly denoted with the variable ''U'' or ''R''.

A matrix that is both upper and lower triangular is [[diagonal matrix|diagonal]]. Matrices that are [[similar (linear algebra)|similar]] to triangular matrices are called '''triangularisable'''.

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]].

===Examples===

The matrix

:<math>\begin{bmatrix}
  1 & 0 & 0 \\
  2 & 96 & 0 \\
  4 & 9 & 69 
\end{bmatrix}</math>

is lower triangular, and

:<math>\begin{bmatrix}
  1 & 4 & 1 \\
  0 & 6 & 9 \\
  0 & 0 & 1 
\end{bmatrix}</math>

is upper triangular.