Upper Triangular Matrix

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What is a Matrix?

A matrix can be defined as a set of numbers that are arranged in rows and columns to create a rectangular array. The numbers in the matrix are known as the elements, or entries, of the matrix. History says that a matrix was not initially known as a matrix but was called the determinant where it was associated with a square array of numbers.

Under certain conditions, we can also add and multiply matrices as individual entities, to give rise to important mathematical systems known as matrix algebras. Two matrices say A and B will be equal to one another if both of them possess an equal number of rows and columns.

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Triangular Matrices

A triangular matrix is a square matrix where all its entries above the principal diagonal or below the principal diagonal are zero. A matrix that has all its entries below the principal diagonal as zero is called the upper triangular matrix. A matrix that has all its entries above the principal diagonal as zero is called the lower triangular matrix.

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The upper triangular matrix can also be called a right triangular matrix and the lower triangular matrix can also be called a left triangular matrix. Apart from these two matrices, there are 3 more special types of matrices. They are named as Unitriangular matrix, Strictly Triangular Matrix, and Atomic Triangular Matrix.

Properties of Upper Triangular Matrices

The important properties of an upper triangular matrix are listed below.

  1. On adding two upper triangular matrices, the result will be an upper triangular matrix itself.

  2. Also, if we multiply two upper triangular matrices, the result will be an upper triangular matrix.

  3. The upper triangular matrix will remain an upper triangular matrix if inversed. 

  4. The transpose of an upper triangular matrix will be a lower triangular matrix, UT = L.

  5. The matrix will remain an upper triangular matrix if it is multiplied to a scalar quantity.

Examples of Upper Triangular Matrix

\[\begin{bmatrix}5 &5  &8 \\0  &3  &10 \\0  &0  &8 \end{bmatrix}\]    \[\begin{bmatrix}-1 &7  &3 \\0  &6  &1 \\0  &0  &5 \end{bmatrix}\]   \[\begin{bmatrix}3 &0  &3 \\0  &7  &-1 \\0  &0  &2 \end{bmatrix}\]

Fun Fact

  1. The term matrix was first introduced by an English mathematician named James Sylvester during the19th-century. But it was his friend, Arthur Cayley, a mathematician who developed the algebraic aspect of matrices. 

  2. Matrices transpire naturally in a system of simultaneous equations.

FAQ (Frequently Asked Questions)

Question 1) What is the Application of Matrices?

Answer 1) The use of matrices in our daily life is much more than anyone can ever imagine. The use or the examples of matrices is always in front of us every day when we go to work or maybe school or university. Given below are some detailed applications of matrices: 

  1. Encryption: In encryption, we use matrices to scramble the data for security purposes, basically to encode or to decode the data. The encoding and decoding of the data can be done with the help of a key that is generated by matrices.

  2. Games Especially 3Ds: Matrices are used to modify or reconstruct the object, in 3d space. They use the 3d matrix to a 2d matrix to switch it into the different objects as per requirement.

  3. Economics and Business: In economics and business studies, a matrix is used to study the trends of a business, shares, to create business models, etc.

  4. Construction: Usually the buildings that we see are straight but sometimes architects construct buildings with a little changed the outer structure, for example, the famous Burj Khalifa, etc. This is done using matrices. We know that a matrix is made of rows and columns. If we change the number of rows and columns within a matrix, we can construct such buildings.

  5. Dance: Matrices are used to structure complicated group dances.

  6. Animation: Matrices can make animations more precise and perfect.

  7. Physics: In physics, we use matrices in the study of electrical circuits, optics, and quantum mechanics. It helps us in the calculation of battery power outputs. With matrices, a resistor conversion of electrical energy into another useful energy is also possible. Therefore, we can say that matrices play a dominant role in calculations especially when it comes to solving the problems using Kirchoff’s laws of voltage and current.

  8. Graphic Software: In applications such as Adobe Photoshop uses matrices to process linear transformations to represent images. 

  9. Geology: Matrices are also helpful in taking seismic surveys.

  10. In hospitals, matrices are used for medical imaging, CAT scans, and MRI’s.

  11. Engineering: Engineers also use matrices for Fourier analysis, Gauss Theorem, to find forces in the bridge, etc. Chemical engineering requires perfectly calibrated computations that are obtained from matrix transformations. 

  12. Other Uses: Matrices are also used in electronics networks, airplanes, and spacecraft.

Question 2) How Many Types of Matrices are There?

Answer 2) There are many different types of matrices. Let us have a look.

The different types of matrices are row and column matrix, zero or null matrix, singleton matrix, vertical and horizontal matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, equal matrix, triangular matrix, singular, and non-singular matrix, symmetric matrix, skew-symmetric matrix, hermitian matrix, skew-hermitian matrix, orthogonal matrix, idempotent matrix, involuntary matrix, and nilpotent matrix.