Triangular Matrix

A triangular matrix is a square matrix in which elements below and/or above the diagonal are all zeros. We have mainly two types of triangular matrices.

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

In this article, let us explore the different types of triangular matrices including upper triangular matrix and lower triangular matrix, their definitions, and their properties. We will also solve some examples based on a triangular matrix for a better understanding of the concept.

A triangular matrix is a special kind of square matrix in the set of matrices. There are two types of triangular matrices: lower triangular matrix and upper triangular matrix.

• A square matrix is said to be a lower triangular matrix if all the elements above its main diagonal are zero.
• A square matrix is said to be an upper triangular matrix if all the elements below the main diagonal are zero.

An example of a triangular matrix is given below:

\(A = \left[\begin{array}{ccc} 2 & -1 & 3 \\ 0 & 5 & 2\\ 0 & 0 & -2 \end{array}\right]\) (upper triangular)

\(B = \left[\begin{array}{ccc} 2 & 0 & 0 \\ 1 & 5 & 0 \\ 1 & -1 & -2 \end{array}\right]\) (lower triangular)

There are different types of triangular matrices that we study. Given below is a list of some special types of triangular matrices:

• Upper Triangular Matrix: A triangular matrix is said to be an upper triangular matrix if all the elements below the main diagonal are zero.
• Lower Triangular Matrix: A triangular matrix is said to be a lower triangular matrix if all the elements above the main diagonal are zero.
• Strictly Triangular Matrix: A triangular matrix is said to be a strictly triangular matrix if all the elements of the main diagonal are zero.
• Strictly Lower Triangular Matrix: A lower triangular matrix is said to be a strictly lower triangular matrix if all the elements of the main diagonal are zero.
• Strictly Upper Triangular Matrix: An upper triangular matrix is said to be a strictly upper triangular matrix if all the elements of the main diagonal are zero.
• 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.
• Unit Lower Triangular Matrix: A lower triangular matrix is said to be a unit lower triangular matrix if all the elements of the main diagonal are equal to 1.
• Unit Upper Triangular Matrix: An upper triangular matrix is said to be a unit upper triangular matrix if all the elements of the main diagonal are equal to 1.

In the following sections, we will mainly explore two types of triangular matrices, namely an upper and lower triangular matrix.

An n × n square matrix A = [a_ij] is said to be an upper triangular matrix if and only if a_ij = 0, for all i > j. This implies that all elements below the main diagonal of a square matrix are zero in an upper triangular matrix. A general notation of an upper triangular matrix is U = [u_ij for i ≤ j, 0 for i > j]. An example of an upper triangular matrix is given below:

\(U = \left[\begin{array}{ccc} 6 & 0 & 8 \\ 0 & 10 & -12\\ 0 & 0 & 2 \end{array}\right]\)

An n × n square matrix A = [a_ij] is said to be a lower triangular matrix if and only if a_ij = 0, for all i < j. This implies that all elements above the main diagonal of a square matrix are zero in a lower triangular matrix. A general notation of a lower triangular matrix is L = [l_ij for i ≥ j, 0 for i < j]. An example of a lower triangular matrix is given below:

\(L = \left[\begin{array}{ccc} -1 & 0 & 0 \\ 0 & -8 & 0 \\ 3 & -4 & 2 \end{array}\right]\)

Since we have understood the meaning of a triangular matrix, let us go through some of its important properties. Given below is a list of the properties of a triangular matrix:

• The transpose of a triangular matrix is triangular.
• The transpose of a lower triangular matrix is n 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.

Important Notes on Triangular Matrix

• An invertible matrix can be written as a product of a lower triangular and upper triangular matrix if and only if its leading principal minors are non-zero. This is also known as the LU decomposition.
• In a matrix is both upper and lower triangular, then it is called as a diagonal matrix.

Related Topics on Triangular Matrix

• Matrix Calculator
• Matrix formula

What is a Triangular Matrix in Linear Algebra?

A triangular matrix is a special type of square matrix in linear algebra whose elements below and above the diagonal appear to be in the form of a triangle. The elements either above and/or below the main diagonal of a triangular matrix are zero.

What are the Properties of a Triangular Matrix?

Some of the important properties of triangular matrices are:

• The transpose of a triangular matrix is triangular.
• The product of two triangular matrices is a triangular matrix.
• The determinant of a triangular matrix is equal to the product of the elements of the main diagonal.

What is a Matrix Called if it is Both Upper and Lower Triangular?

If a matrix is both lower and upper triangular, then all its non-diagonal elements are equal to zeros. In this case, it is called a diagonal matrix.

When is a Triangular Matrix Invertible?

A triangular matrix (lower or upper) is invertible if and only if all entries of the main diagonal are non-zero.

What is an Upper Triangular Matrix?

An n × n square matrix A = [a_ij] is said to be an upper triangular matrix if and only if a_ij = 0, for all i > j. This implies that all elements below the main diagonal of a square matrix are zero in an upper triangular matrix.

What is the Inverse of a Lower Triangular Matrix?

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

How to Find the Determinant of a Triangular Matrix?

The determinant of a triangular matrix can be found by taking the product of the elements of the main diagonal.

What are the Eigen Values of a Triangular Matrix?

The eigenvalues of a triangular matrix (upper or lower) are the elements of the main diagonal of the triangular matrix.