A triangular matrix is a type of square matrix used in linear algebra where the numbers above or below the main diagonal are arranged in the shape of a triangle. A matrix itself is simply a set of numbers written in rows and columns. There are different types of matrices such as row matrices, column matrices, rectangular matrices, diagonal matrices, zero matrices, identity matrices, and triangular matrices. In a triangular matrix, either the lower or the upper part is filled with zeros, which makes it easier to solve equations and perform calculations in mathematics.

What is a Triangular Matrix?

A triangular matrix is a special type of square matrix where all the values above or below the diagonal are zero. The triangular matrix can be classified into two types which are given below:

• Upper Triangular Matrix: A square matrix whose all elements below the main diagonal are zero is called an upper triangular matrix.
• Lower Triangular Matrix: A square matrix whose all elements above the main diagonal are zero is called a lower triangular matrix.

 Let us explore the different types of triangular matrices including upper triangular matrix and lower triangular matrix, their definitions, their types, and their properties.

What is the Upper Triangular Matrix?

An upper triangular matrix can be defined as a square matrix in which all the elements below the main diagonal are zero. In other words, an \(n \times n\) square matrix \(A=[a_{ij}]\) is called an upper triangular matrix if and only if \(a_{ij} = 0\), for all \(i>j\).

\(U=\left\{\begin{matrix}u_{ij}&:i\leq j\\0&:i>j \\\end{matrix}\right.\) is a common notation for an upper triangular matrix.

 
Upper Triangular Matrix Examples

Below are a few examples of upper triangular matrix:

• \(\begin{bmatrix}1&-1\\0&2\\\end{bmatrix}\) is an \(2 \times 2\) upper triangular matrix.

• \(\begin{bmatrix}1&2&4\\0&3&5\\0&0&6\\\end{bmatrix}\) is an \(3 \times 3\) upper triangular matrix.
• \(\begin{bmatrix}3&2&6&7\\0&6&6&8\\0&0&2&4\\0&0&0&1\\\end{bmatrix}\) is an \(4 \times 4\) upper triangular matrix.

What is the Lower Triangular Matrix?

A lower triangular matrix can be defined as a square matrix in which all the elements above the main diagonal are zero. In other words, an \(n \times n\) square matrix \(A=[a_{ij}]\) is called a lower triangular matrix if and only if \(a_{ij} = 0\), for all \(i<j\).

\(L=\left\{\begin{matrix}l_{ij}&:i\geq j\\0&:i<j \\\end{matrix}\right. \) is a common notation for a lower triangular matrix.

Lower Triangular Matrix Examples

Below are a few examples of lower triangular matrix:

• \(\begin{bmatrix}1&0\\-5&2\\\end{bmatrix}\) is an \(2 \times 2\) lower triangular matrix.

• \(\begin{bmatrix}-1&0&0\\0&-8&0\\3&-4&2\\\end{bmatrix}\) is an \(3 \times 3\) lower triangular matrix.
• \(\begin{bmatrix}3&0&0&0\\-1&6&0&0\\5&0&-2&0\\7&3&-2&1\\\end{bmatrix}\) is an \(4 \times 4\) lower triangular matrix.

Types of Triangular Matrix

There are different types of triangular matrices that we will learn in this section. Given below is a list of some special types of triangular matrices:

Unit Triangular Matrix: A triangular matrix is said to be a unit triangular matrix if all the elements of the main diagonal are equal to \(1\). A unit triangular matrix can be further classified into unit upper triangular and unit lower triangular matrix.

Strictly Triangular Matrix: A triangular matrix is said to be a strictly triangular matrix if all the elements of the main diagonal are zero. A strictly triangular matrix can be further classified into strictly upper triangular and strictly lower triangular matrix.

Strictly Upper Triangular Matrix: A strictly upper triangular matrix has zero entries on the downward diagonal and non-zero entries above it.

Unit Upper Triangular Matrix: A unit upper triangular matrix is a matrix that has \(1\) as entries on the downwards diagonal and non-zero entries above it.

Strictly Lower Triangular Matrix: A strictly lower triangular matrix has zero entries on the downwards diagonal and non-zero entries below it.

Unit Lower Triangular Matrix: A unit lower triangular matrix is a matrix that has \(1\) as all entries on the downwards diagonal and non-zero entries below it.

How to Convert a Matrix to Upper Triangular Matrix

Let us learn how to convert a matrix to an upper triangular matrix with examples.

Example: Transform \(A=\begin{bmatrix}1&-1&2\\2&1&3\\3&2&4\\\end{bmatrix}\) into an upper triangular matrix by suitable row transformation.

Solution: Given \(A=\begin{bmatrix}1&-1&2\\2&1&3\\3&2&4\\\end{bmatrix}\)

Apply row transformation, \(R_{2}\to R_{2}-2R_{1}\)

\(\Rightarrow\) \(A=\begin{bmatrix}1&-1&2\\0&3&1\\3&2&4\\\end{bmatrix}\)

\(R_{3}\to R_{3}-3R_{1}\)

\(\Rightarrow\) \(A=\begin{bmatrix}1&-1&2\\0&3&1\\0&5&-2\\\end{bmatrix}\)

\(R_{3}\to R_{3}-\frac{5}{3}R_{2}\)

\(\Rightarrow\) \(A=\begin{bmatrix}1&-1&2\\0&3&1\\0&0&-\frac{1}{3}\\\end{bmatrix}\)

Thus, the matrix \(A=\begin{bmatrix}1&-1&2\\0&3&1\\0&0&-\frac{1}{3}\\\end{bmatrix}\) is now an upper triangular matrix.

How to Convert a Matrix to Lower Triangular Matrix

Let us learn how to convert a matrix to a lower triangular matrix with examples.

Example: Transform \(A=\begin{bmatrix}2&1&3\\7&5&2\\3&2&1\\\end{bmatrix}\) into a lower triangular matrix by suitable row transformation.

Solution: Given \(A=\begin{bmatrix}2&1&3\\7&5&2\\3&2&1\\\end{bmatrix}\)

Apply row transformation, \(R_{2}\to R_{2}-2R_{3}\)

\(\Rightarrow\) \(A=\begin{bmatrix}2&1&3\\1&1&0\\3&2&1\\\end{bmatrix}\)

\(R_{1}\to R_{1}-3R_{3}\)

\(\Rightarrow\) \(A=\begin{bmatrix}-7&-5&0\\1&1&0\\3&2&1\\\end{bmatrix}\)

\(R_{1}\to R_{1}+5R_{2}\)

\(\Rightarrow\) \(A=\begin{bmatrix}-2&0&0\\1&1&0\\3&2&1\\\end{bmatrix}\)

Thus, the matrix \(A=\begin{bmatrix}-2&0&0\\1&1&0\\3&2&1\\\end{bmatrix}\) is now a lower triangular matrix.

Learn about Determinant of 4 x 4 Matrix

Determinant of Triangular Matrix

The determinant of a triangular matrix is the product of the elements of the main diagonal. Now let us discuss the determinant of the upper triangular and lower triangular matrix with examples.

Determinant of Upper Triangular Matrix

The determinant of the upper triangular matrix is the product of the main diagonal entries of the upper triangular matrix.

Example: Find the determinant of the upper triangular matrix \(A=\begin{bmatrix}1&4&2\\0&3&4\\0&0&1\\\end{bmatrix}\).

Solution: The given matrix is an upper triangular matrix as its elements below the diagonal are all zeros.

Hence its determinant is the product of its diagonal elements.

\(\Rightarrow\) \(det(A) = (1)(3)(1) = 3\).

Therefore, the determinant of the given upper triangular matrix is \(3\).

Determinant of Lower Triangular Matrix

The determinant of the lower triangular matrix is the product of the main diagonal entries of the lower triangular matrix.

Example: Find the determinant of the lower triangular matrix \(A=\begin{bmatrix}2&0&0\\5&1&0\\6&7&3\\\end{bmatrix}\).

Solution: The given matrix is a lower triangular matrix as its elements above the diagonal are all zeros.

Hence its determinant is the product of its diagonal elements.

\(\Rightarrow\) \(det(A) = (2)(1)(3) = 6\).

Therefore, the determinant of the given lower triangular matrix is \(6\).

Inverse of Triangular Matrix

The inverse of a triangular matrix is also a triangular matrix. Inverse exists only if none of the diagonal elements is zero. Now let us discuss the inverse of an upper triangular and lower triangular matrix with examples.

Inverse of Upper Triangular Matrix: The inverse of an upper triangular matrix is also an upper triangular matrix.

Example: Find the inverse of the upper triangular matrix \(A=\begin{bmatrix}2&1\\0&-1\\\end{bmatrix}\).

Solution: The determinant of the matrix \(|A| = -2 \neq 0\), this implies the inverse of the given upper triangular matrix exists.

Now we know, \(A^{-1}=\frac{1}{|A|}\begin{bmatrix}-1&-1\\0&2\\\end{bmatrix}\)

\(\Rightarrow\) \(A^{-1}=\frac{1}{(-2)}\begin{bmatrix}-1&-1\\0&2\\\end{bmatrix}\)

\(\Rightarrow\) \(A^{-1}=\begin{bmatrix}0.5&0.5\\0&-1\\\end{bmatrix}\)

Thus, we can see that the inverse of an upper triangular matrix is an upper triangular matrix.

Inverse of Lower Triangular Matrix: The inverse of a lower triangular matrix is also a lower triangular matrix.

Example: Find the inverse of the upper triangular matrix \(A=\begin{bmatrix}2&0\\2&1\\\end{bmatrix}\).

Solution: The determinant of the matrix \(|A| = 2 \neq 0\), this implies the inverse of the given lower triangular matrix exists.

Now we know, \(A^{-1}=\frac{1}{|A|}\begin{bmatrix}1&0\\-2&2\\\end{bmatrix}\)

\(\Rightarrow\) \(A^{-1}=\frac{1}{2}\begin{bmatrix}1&0\\-2&2\\\end{bmatrix}\)

\(\Rightarrow\) \(A^{-1}=\begin{bmatrix}0.5&0\\-1&1\\\end{bmatrix}\)

Thus, we can see that the inverse of a lower triangular matrix is a lower triangular matrix.

Eigenvalue of Triangular Matrix

The eigenvalues of a triangular matrix are the diagonal entries of the matrix. Now let us discuss the eigenvalues of the upper triangular and lower triangular matrix with examples.

Eigenvalue of Upper Triangular Matrix: The eigenvalues of the upper triangular matrix are the diagonal entries of the upper triangular matrix.

Example: Find the eigenvalues of the upper triangular matrix \(A=\begin{bmatrix}1&4&2\\0&3&4\\0&0&5\\\end{bmatrix}\).

Solution: The given matrix is an upper triangular matrix as its elements below the diagonal are all zeros.

Hence its eigenvalues are the diagonal elements.

The diagonal elements are \(1\), \(3\), and \(5\).

Therefore, the eigenvalues of the given upper triangular matrix are \(1\), \(3\), and \(5\).

Eigenvalue of Lower Triangular Matrix: The eigenvalues of the lower triangular matrix are the diagonal entries of the lower triangular matrix.

Example: Find the eigenvalues of the lower triangular matrix \(A=\begin{bmatrix}7&0&0\\5&4&0\\6&1&3\\\end{bmatrix}\).

Solution: The given matrix is a lower triangular matrix as its elements above the diagonal are all zeros.

Hence its eigenvalues are the diagonal elements.

The diagonal elements are \(7\), \(4\), and \(3\).

Therefore, the eigenvalues of the given lower triangular matrix are \(7\), \(4\), and \(3\).

Triangular Matrix Properties

Some of the important properties of the triangular matrix are listed below:

• The transpose of a triangular matrix is triangular.
• The transpose of a lower triangular matrix is an upper triangular matrix and vice-versa.
• The product of two triangular matrices is a triangular matrix.
• A triangular matrix is invertible if and only if all entries of the main diagonal are non-zero.
• The product of two lower(upper) triangular matrices is a lower(upper) triangular matrix.
• The inverse of a triangular matrix is triangular.
• The determinant of a triangular matrix is the product of the elements of the main diagonal.
• The eigenvalues of a triangular matrix are the diagonal entries of the matrix.

Applications of Upper & Lower Triangular Matrices

Upper and lower triangular matrices are very useful in mathematics, computer science, and real-life problem solving. These matrices make calculations faster and easier because many entries are zero. Below are some important applications explained in simple words

1. Solving Systems of Equations (LU Decomposition)

Any invertible square matrix can often be written as the product of a lower triangular matrix (L) and an upper triangular matrix (U).This process is called LU decomposition. It is widely used to solve linear equations quickly, especially when the system is large.

1. Finding Determinants Easily

The determinant of a triangular matrix is simply the product of the numbers on its diagonal.This makes it much faster to calculate compared to other matrices.

• Inverse of Matrices

If a matrix is an invertible triangular matrix, the inverse of this matrix is also triangular. This makes it easier to simplify the process of finding matrix inverses.

• In Computer Algorithms

Triangular matrices are employed in computer algorithms and numerical methods.They minimize the number of calculations that are required, and this saves time and memory.

• Eigenvalues

The eigenvalues of a triangular matrix are simply the numbers along the main diagonal.This characteristic makes triangular matrices highly applicable to linear algebra and physics.

• Engineering and Data Science

In engineering problems, triangular matrices are employed for stability tests, control systems, and simulations.In data science and machine learning, they emerge in optimisation and matrix factorisation techniques.

Differences Between Upper & Lower Triangular Matrix

An upper triangular matrix has all zero elements below the main diagonal, while a lower triangular matrix has all zero elements above the main diagonal. The key difference is in the position of the zero values around the diagonal.

Important Notes on Triangular Matrix

1. LU Decomposition:
   Any matrix that can be inverted (non-singular matrix) can sometimes be written as the product of two special matrices – a lower triangular matrix and an upper triangular matrix. This process is called LU decomposition. For this to be possible, the matrix must have all its leading principal minors (determinants of top-left submatrices) not equal to zero.
2. Diagonal Matrix:
   If a matrix is both an upper triangular matrix (all elements below the main diagonal are zero) and a lower triangular matrix (all elements above the main diagonal are zero), then the matrix is a diagonal matrix. This means that only the numbers on the main diagonal are non-zero, while every other entry is zero.

Solved Examples of Triangular Matrix

Example 1: Classify the following matrices into upper and lower triangular matrices:

i). \(\begin{bmatrix}1&4&2\\0&3&4\\0&0&1\\\end{bmatrix}\)

ii). \(\begin{bmatrix}1&0&0\\2&8&0\\4&9&7\\\end{bmatrix}\)

iii). \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&3\\\end{bmatrix}\)

iv). \(\begin{bmatrix}2&3&5\\0&8&9\\0&0&7\\\end{bmatrix}\)

v). \(\begin{bmatrix}1&2&3&4\\0&0&3&4\\0&0&0&5\\\end{bmatrix}\)

Solution:

i). \(\begin{bmatrix}1&4&2\\0&3&4\\0&0&1\\\end{bmatrix}\): Upper triangular matrix.

ii). \(\begin{bmatrix}1&0&0\\2&8&0\\4&9&7\\\end{bmatrix}\): Lower triangular matrix.

iii). \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&3\\\end{bmatrix}\): Lower as well as upper triangular matrix.

iv). \(\begin{bmatrix}2&3&5\\0&8&9\\0&0&7\\\end{bmatrix}\): Upper triangular matrix.

v). \(\begin{bmatrix}1&2&3&4\\0&0&3&4\\0&0&0&5\\\end{bmatrix}\): Neither upper triangular nor lower triangular matrix because it is not a square matrix.

Example 2: Find the determinant of the matrix \(A=\begin{bmatrix}1&0&0\\2&8&0\\4&9&7\\\end{bmatrix}\).

Solution: The given matrix is a lower triangular matrix as its elements above the diagonal are all zeros.

Hence its determinant is the product of its diagonal elements.

\(\Rightarrow\) \(det(A) = (1)(8)(7) = 56\).

Therefore, the determinant of the given matrix \(A\) is \(56\).

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