Triangular Matrices

Upper and Lower Triangular Matrices

Definition: A square matrix $A$ is said to be an Upper Triangular Matrix if all entries below the main diagonal are zero (if $i > j$, $a_{ij} = 0$) and called a Lower Triangular Matrix if all entries above the main diagonal are zero (if $i < j$, $a_{ij} = 0$).

An easy way to remember whether a matrix is upper triangular or lower triangular by where the non-zero entries of the matrix lie as illustrated in the following graphic:

For example, matrix $A$ is an upper triangular matrix while matrix $B$ is a lower triangular matrix:

(1)

\begin{align} A = \begin{bmatrix} 1 & 2 & 4\\ 0 & 3 & \frac{1}{2}\\ 0 & 0 & 3 \end{bmatrix} \quad , \quad B = \begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 0\\ 1 & 1 & -2 \end{bmatrix} \end{align}

Note that zeroes can appear elsewhere in triangular matrices as in matrix $B$ where entry $b_{31} = 0$.

Theorem 1: Let $A$ and $B$ be square $n \times n$ triangular matrices. Then: a) If $A$ is an upper triangular matrix, then $A^T$ is a lower triangular matrix. If $A$ is a lower triangular matrix, then $A^T$ is an upper triangular matrix. b) If $A$ and $B$ are both upper triangular matrices, then $AB$ is an upper triangular matrix. If $A$ and $B$ are both lower triangular matrices, then $AB$ is a lower triangular matrix.

• Proof of (a): If $A$ is an upper triangular matrix, transposing A results in "reflecting" entries over the main diagonal. Therefore the triangle of zeroes in the bottom left corner of $A$ will be in the top right corner of $A^T$. Since $A$ is square, so is $A^T$, so $A^T$ must be a lower triangular matrix. The same intuition works for lower triangular matrices and their transposes. $\blacksquare$

We will omit the proof of part (b) as it similar to that of (a).