Triangular Matrix

Let $n$ be a positive integer.

An upper triangular matrix is of the form:

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & a_{22} & a_{23} & \cdots & a_{2n} \\ 0 & 0 & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_{nn} \end{bmatrix} $$

An upper triangular matrix is sometimes also called right triangular.

A lower triangular matrix is of the form:

$$ \begin{bmatrix} a_{11} & 0 & 0 & \cdots & 0 \\ a_{21} & a_{22} & 0 & \cdots & 0 \\ a_{31} & a_{32} & a_{33} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \end{bmatrix} $$

A lower triangular matrix is sometimes also called left triangular.

Note that upper triangular matrices and lower triangular matrices must be square matrices.

A triangular matrix is a matrix that is an upper triangular matrix or lower triangular matrix. Note that some matrices, such as the identity matrix, are both upper and lower triangular. A matrix is upper and lower triangular simultaneously if and only if it is a diagonal matrix.

Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, if $A$ is an $n\times n$ triangular matrix, the equation $Ax=b$ can be solved for $x$ in at most $n^2$ operations.

In fact, triangular matrices are so useful that much computational linear algebra begins with factoring (or decomposing) a general matrix or matrices into triangular form. Some matrix factorization methods are the Cholesky factorization and the LU-factorization. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.

Properties

Triangular matrices have the following properties (prefix ``triangular'' with either ``upper'' or ``lower'' uniformly):

• The inverse of a triangular matrix is a triangular matrix.
• The product of two triangular matrices is a triangular matrix.
• The determinant of a triangular matrix is the product of the diagonal elements.
• The eigenvalues of a triangular matrix are the diagonal elements.

The last two properties follow easily from the cofactor expansion of the triangular matrix.