Triangular Matrix

What is a Triangular Matrix?

A triangular matrix is a special type of square matrix in linear algebra, where the arrangement of elements depends on the position of the main diagonal. It is widely used in solving linear equations, matrix factorization, and determinant problems.

Definition of Triangular Matrix

A triangular matrix is defined as a square matrix ($n \times n$) in which either all elements above or all elements below the main diagonal are zero.

• If $a_{ij} = 0$ for all $i > j$, the matrix is called an upper triangular matrix.

• If $a_{ij} = 0$ for all $i < j$, the matrix is called a lower triangular matrix.

Example of an upper triangular matrix:

$A = \begin{bmatrix} 2 & 5 & 3 \\ 0 & 4 & 6 \\ 0 & 0 & 7 \end{bmatrix}$

Here, all entries below the main diagonal are zero.

Main Diagonal and Its Role in Triangular Matrices

The main diagonal of a matrix is formed by elements where $i = j$. In a triangular matrix, the main diagonal often contains the key non-zero values.

For example, in $B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & 9 \end{bmatrix}$

the main diagonal elements are $1, 5, 9$.

The main diagonal plays an important role because:

• The determinant of a triangular matrix is the product of its diagonal elements.

• Eigenvalues of triangular matrices are always equal to the diagonal elements.

A triangular matrix is further classified into two types:

1. Upper triangular matrix
2. Lower triangular matrix

Types of Triangular Matrices

Triangular matrices are broadly divided into two types of matrices namely as upper triangular and lower triangular matrices, depending on the position of zeros with respect to the main diagonal.

Upper Triangular Matrix – Definition and Example

An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.

Mathematically, if $a_{ij} = 0$ for all $i > j$, then $A$ is an upper triangular matrix.

An Upper triangular matrix is denoted by Letter ‘U’

$\begin{aligned} & \text { Or } \mathrm{A}=\left[\mathrm{a}_{\mathrm{i} \mathrm{j}}\right]_{\mathrm{m} \times \mathrm{n}} \text { is said to be upper triangular if } \mathrm{A}=\left[\mathrm{a}_{\mathrm{i} \mathrm{j}}\right]_{\mathrm{m} \times \mathrm{n}}=0 \text { when } \mathrm{i}>\mathrm{j} \text {. } \\ & \qquad\left[\begin{array}{ccccc}a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ 0 & a_{22} & a_{23} & a_{24} & a_{25} \\ 0 & 0 & a_{33} & a_{34} & a_{35} \\ 0 & 0 & 0 & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55}\end{array}\right]\end{aligned}$

Properties of Upper triangular matrix

Numerous operations preserve upper triangularity:

• Upper triangular is the product of two upper triangular matrices.
• Upper triangular is the result of multiplying two upper triangular matrices.
• If an upper triangular matrix exists, its inverse is also upper triangular.
• Upper triangular is the result of multiplying a scalar by an upper triangular matrix.

Example: $A = \begin{bmatrix} 3 & 4 & 7 \\ 0 & 5 & 6 \\ 0 & 0 & 8 \end{bmatrix}$

Here, entries below the diagonal ($a_{21}, a_{31}, a_{32}$) are all zero.

Lower Triangular Matrix – Definition and Example

A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.

If $a_{ij} = 0$ for all $i < j$, then $B$ is a lower triangular matrix.

The Lower triangular matrix is denoted by ‘L’

Or $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be upper triangular if $\mathrm{A}=\left[\mathrm{a}_{\mathrm{i} j}\right]_{\mathrm{m} \times \mathrm{n}}=0$ when $\mathrm{i}<\mathrm{j}$.

Example,
$
\text {}\left[\begin{array}{ccccc}
a_{11} & 0 & 0 & 0 & 0 \\
a_{21} & a_{22} & 0 & 0 & 0 \\
a_{31} & a_{32} & a_{33} & 0 & 0 \\
a_{41} & a_{42} & a_{43} & a_{44} & 0 \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}\right]
$

Properties of Lower triangular matrix

Numerous operations preserve lower triangularity:

• Lower triangular is the product of two upper triangular matrices.
• Lower triangular is the result of multiplying two upper triangular matrices.
• If a Lower triangular matrix exists, its inverse is also upper triangular.
• Lower triangular is the result of multiplying a scalar by an upper triangular matrix.

Example: $B = \begin{bmatrix} 2 & 0 & 0 \\ 5 & 3 & 0 \\ 7 & 4 & 6 \end{bmatrix}$

Here, entries above the diagonal ($a_{12}, a_{13}, a_{23}$) are zero.

How to Identify a Triangular Matrix

Identifying whether a matrix is triangular or not is simple if you check the position of zeros with respect to the main diagonal.

Step-by-Step Method to Check Triangular Form

1. Write down the matrix clearly.

2. Look at the main diagonal elements ($a_{11}, a_{22}, a_{33}, \dots$).

3. Check the entries below the diagonal:

   • If all are zero ($a_{ij} = 0$ for $i > j$), then it is an upper triangular matrix.

4. Check the entries above the diagonal:

   • If all are zero ($a_{ij} = 0$ for $i < j$), then it is a lower triangular matrix.

5. If neither condition holds, the matrix is not triangular.

Example:

$A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}$ → Upper triangular

$B = \begin{bmatrix} 7 & 0 & 0 \\ 8 & 5 & 0 \\ 9 & 2 & 3 \end{bmatrix}$ → Lower triangular

Difference Between Upper and Lower Triangular Matrices

The difference between upper and lower triangular matrices lies in the placement of zero elements relative to the main diagonal. Below is a simple comparison of their key properties.

• In an upper triangular matrix, non-zero values appear on and above the diagonal, while in a lower triangular matrix, they appear on and below the diagonal.

• Example of upper: $ \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} $

• Example of lower: $ \begin{bmatrix} 1 & 0 \\ 4 & 3 \end{bmatrix} $

Special Forms of Triangular Matrix

Special forms of triangular matrices include unit triangular and strictly triangular matrices, which have unique patterns on their diagonal elements. Below is a brief overview of these special forms.

Unit Triangular Matrix

An upper or lower triangular matrix is called a unit triangular matrix if all elements on its main diagonal are equal to 1.
For example, in an $n \times n$ matrix $A = [a_{ij}]$, if $a_{ii} = 1$ for all $i$, then $A$ is a unit triangular matrix.

Other terms used are unit (upper or lower) triangular or, rarely, normed triangular matrices. It is important to note that:

• A unit triangular matrix is not the same as a unit matrix (identity matrix).

• A normed triangular matrix has no relation to the concept of matrix norms.

Strictly Triangular Matrix

A strictly triangular matrix is defined as a triangular matrix in which all the entries on the main diagonal are zero.
That is, for a matrix $A = [a_{ij}]$, if $a_{ii} = 0$ for all $i$, then $A$ is strictly upper or strictly lower triangular.

According to the Cayley-Hamilton theorem, any finite strictly triangular matrix is nilpotent, with nilpotency index at most $n$ (where $n$ is the order of the matrix).

Properties of Triangular Matrices

Triangular matrices have some important properties that make them useful in linear algebra, determinants, and matrix factorisation.

Determinant of a Triangular Matrix

The determinant of a triangular matrix is always equal to the product of its diagonal elements.

If $A = \begin{bmatrix} a_{11} & * & * \\ 0 & a_{22} & * \\ 0 & 0 & a_{33} \end{bmatrix}$

then

$\det(A) = a_{11} \cdot a_{22} \cdot a_{33}$

This property holds for both upper and lower triangular matrices.

Inverse of a Triangular Matrix

• The inverse of an upper triangular matrix (if it exists) is also an upper triangular matrix.

• The inverse of a lower triangular matrix is also a lower triangular matrix.

• In both cases, the diagonal elements must be non-zero for the inverse to exist.

Example: If $A = \begin{bmatrix} 2 & 3 \\ 0 & 4 \end{bmatrix}$

then $A^{-1}$ will also be upper triangular.

1. The transpose of a lower triangular matrix is an upper triangular matrix, and vice versa:

   • $(U^T = L)$, $(L^T = U)$

2. The determinant of a triangular matrix is equal to the product of its diagonal elements:

   • $\det(A) = \prod_{i=1}^n a_{ii}$

3. A triangular matrix is invertible if and only if all its diagonal elements are non-zero.

4. The product of two triangular matrices of the same type is also triangular:

   • If $U_1$ and $U_2$ are upper triangular, then $U_1U_2$ is also upper triangular.

   • If $L_1$ and $L_2$ are lower triangular, then $L_1L_2$ is also lower triangular.

5. When two triangular matrices are added, the resulting matrix remains triangular (upper or lower depending on the type).

Solved Examples Based On Triangular Matrices

Example 1: If A is a Lower triangular matrix with the definition

$\begin{aligned} a_{i j} & =\{i-j ; \text { when } i>j \\ & =\{i+j ; \text { when } i=j \\ & =\{0 ; \text { when } i<j\end{aligned}$

Solution: Lower Triangular Matrix -A square matrix in which all the elements above the principal diagonal are Zero

$a_{i j}=0, i<j$

$\begin{aligned} & a_{11}=1+1=2 ; a_{22}=2+2=4 ; a_{33}=3+3=6 \\ & |A|=a_1 \times a_{22} \times a_{33}=2 \times 4 \times 6=48\end{aligned}$

Hence the value of |A| =48

Example 2: If A is a strictly triangular matrix of order 3 x 3 and $B=\operatorname{diag}\left[\begin{array}{lll}3 & 5 & 2\end{array}\right]$ ; Then |AB|=

1)30

2)5

3) 0

4)28