Upper Triangular Matrix : Definition, Properties, Examples & Application

Upper Triangular Matrix Overview

Triangular Matrix is a type of square matrix in Linear Algebra in which the entries below and above the diagonal look to form a triangle. Triangular matrices can be divided into two types. In a lower triangle matrix, all elements above the main diagonal are zero, whereas in an upper triangular matrix, all elements below the main diagonal are zero. We can calculate the determinant for any upper triangular matrix by multiplying all of its components along the major diagonal. This also implies that if a 0 occurs anywhere along the main diagonal of an upper triangular matrix, the determinant will be 0.

Define Triangular Matrix

A triangular matrix is a type of square matrix in the collection of matrices. There are two kinds of triangle matrices: lower triangular matrices and upper triangular matrices.

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

Below is an example of a triangular matrix:

What is the Upper Triangular Matrix?

An n × n square matrix A = [aij] is called an upper triangular matrix if and only if aij = 0, for all i > j. In an upper triangular matrix, this means that all elements below the major diagonal of a square matrix are zero. U = [uij for I j, 0 for I > j] is a common notation for an upper triangular matrix. Here is an example of an upper triangular matrix:

An upper triangular matrix is one with zero entries below the main diagonal, while a lower triangular matrix has zero entries above the main diagonal. The Upper triangular sparse matrix contains no elements below the main diagonal. Another term for this type of sparse matrix is an upper triangular matrix. When you look at it graphically, you'll notice that all of the components with non-zero values are shown above the diagonal.

Apart from these two, there are some unique form matrices, such as the following:

1. Unitriangular Matrix is a type of unit triangular matrix.
2. Strictly Matrix with three triangles
3. Atomic Matrix with three triangles

Types of Upper Triangular Matrix

There are two main types of upper triangular matrices: strict upper triangular matrices and non-strict (or semi-strict) upper triangular matrices.

1. Strict Upper Triangular Matrix: A strict upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero and all the diagonal entries are also zero. In other words, all the elements below the main diagonal are zero, and none of the elements on the diagonal are non-zero. For example:  0 2 5 0 0 7 0 0 0
2. Non-Strict Upper Triangular Matrix: A non-strict upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero, but some or all of the diagonal entries may be nonzero. In other words, all the elements below the main diagonal are zero, but some or all of the elements on the diagonal are also nonzero. For example: 3 2 5 0 4 7 0 0 1

In both cases, the elements above the main diagonal may be nonzero or zero, and the matrix may have any dimension (i.e., it may be a 2x2 matrix, a 3x3 matrix, etc.). However, the main difference between the two types of upper triangular matrices is the condition of the diagonal entries: strict upper triangular matrices have zero diagonal entries, while non-strict upper triangular matrices may have nonzero diagonal entries.

Properties of Upper Triangular Matrix

A list of the most important properties of an upper triangular matrix is given below.

1. An upper triangular matrix is produced when two upper triangular matrices are combined together.
2. Additionally, an upper triangular matrix is produced when two upper triangular matrices are compounded.
3. The upper triangular matrix will still be an upper triangular matrix even if it is inverted.
4. UT = L signifies the transposition of an upper triangular matrix into a lower triangular matrix.
5. The matrix will continue to be an upper triangular matrix even after being multiplied by a scalar number.
6. In an upper triangular matrix, at least one entry must be non-zero and above the main diagonal. There must be at least one non-zero component in a lower triangular matrix below the main diagonal.
7. An upper triangular matrix is typically denoted by the letter U, whereas a lower triangular matrix is usually denoted by the letter L.

Read more about: Scalar Matrix.

Application of Upper Triangular Matrix

Upper triangular matrices have a wide range of applications in various fields of mathematics, science, and engineering. Here are some of the most common applications of the upper triangular matrix:

1. Encryption: In encryption, matrices are used to jumble data for security reasons, essentially to encode or decode the data. The data can be encoded and decoded with the use of a key that is generated using matrices.
2. 3D Games: Matrices are used to edit or recreate objects in 3D space, especially in games. They convert a 33Dmatrix to a 22Dmatrix to convert it into the many items required.
3. Business and Economics: A matrix is used in economics and business studies to investigate business trends, and share, and construct business models, among other things.
4. Construction: Most structures we see are straight, however, architects occasionally construct skyscrapers with slight variations in the outside construction, such as the iconic Burj Khalifa. Matrixes are used to do this. A matrix is made up of rows and columns, as we all know. We can make such structures by changing the number of rows and columns in a matrix.
5. Engineering: Engineers utilize matrices for Fourier analysis, Gauss Theorem, and finding forces in bridges, among other things. Matrix transformations are used to obtain precisely calibrated computations in chemical engineering.
6. Miscellaneous: Matrices can also be found in electrical networks, planes, and spacecraft.
7. Dance: Matrices are used in dance to structure complex group dances.
8. Animation: Matrices can improve the precision and perfection of animations.
9. Physics: Matrixes are used in the study of electrical circuits, optics, and quantum mechanics in physics. It aids us in determining battery power outputs. It is also feasible to convert electrical energy into other usable energy using matrices. As a result, we can conclude that matrices play a significant role in calculations, particularly when solving problems involving Kirchoff's voltage and current laws.
10. Graphics Software : Matrixes are used in apps like Adobe Photoshop to process linear transformations and represent images.
11. Geology : Matrices are also useful in seismic surveys in geology.
12. Hospital: Matrixes are utilized in hospitals for medical imaging, CAT scans, and MRIs.

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Points to Remember

1. A triangular matrix is a subset of square matrices in which the components below and/or above the diagonal are all zeros. 
2. A triangular matrix can be divided into two types: a lower triangular matrix and a higher triangular matrix. 
3. A lower triangle matrix has all elements above the main diagonal that are zero, whereas an upper triangular matrix has all elements below the main diagonal that are zero.
4. An n n square matrix A = [aij] is said to be an upper triangular matrix if and only if aij = 0, for all i > j. 
5. The general notation for an upper triangular matrix is U = [uij for i j, 0 for i > j].