## Type of matrices in Discrete mathematicsWe will discuss little bit about matrices because we are going to discuss about the types of matrices in this section. A matrix is used to contain the rectangular array of numbers, and all these numbers, expressions, or symbols will be arranged in the form of rows and columns. With the help of a number of rows and columns, we can determine the order of matrix. If there is more than one matrix, then it will be known as the matrices. In any matrix, the entries are known as the numbers, and each number is an element. We can refer to the size of the matrix as m by n, and we can write it as m∗n. Here n is used to indicate the number of rows, and m is used to indicate the number of columns. Discrete mathematics contains many types of matrices. Based on the order, element, and certain set of conditions of the matrices, we can differentiate all these types of matrices. The word "matrices" is used to indicate the plural form of a matrix. We will use the term in this section to indicate more than one matrix. In this section, we are going to learn about different types of matrices, their definitions, and many examples. ## What are Types of matricesHere we will show some important types of matrices which are used in the field of discrete mathematics, science, and engineering. The various types of matrices in discrete mathematics are described as follows: - Row and column matrix
- Rectangular matrix and square matrix
- Horizontal matrix and vertical matrix
- Singleton matrix
- Identity matrix
- Zero matrix
- Diagonal matrix
- Singular matrix and on singular matrix
- Hermitian matrix and Skew Hermitian matrix
- Upper and lower triangular matrix
- Symmetric matrix and Skew symmetric matrix
- Orthogonal matrix
With the help of above types of matrices, we can organize data on the basis of age, month, person, group, company, and many more things. By using this information, we can make decisions and solve various kinds of math problems. ## Identifying types of matricesThere are various sizes in which the matrices can be represented, but usually, the shape of the matrices remains the same. The size of the matrix will be known as its order. We can calculate it with the help of counting the total number of rows and columns in a matrix. The following image will show the procedure for finding the dimension of a given matrix. Now we will learn about the most commonly used matrices and how we can find them on the basis of their dimension. ## Row and column matrixA matrix will be known as a Similarly, a matrix will be known as a
## Rectangular and Square matrixA matrix will be known as a Similarly, a matrix will be known as a
## Identify and Zero matricesA matrix will be known as the ## Note:- If there are identity matrices, then they must be scalar matrices.
- If there are scalar matrices, then they must be diagonal matrices.
- If there are diagonal matrices, then they must be square matrices.
- It is not possible to do the converse of all these statements.
A matrix will be known as a
## Equal matrixThe matrices will be known as Suppose there are two matrices, A and B. These matrices will be equal if and only if the order and the corresponding elements of these matrices are similar to each other. Suppose A = [aij] - If m = r. It shows that the number of rows in matrix A is equal to the number of rows in matrix B.
- If n = s. It shows that the number of columns in matrix A is equal to the number of columns in matrix B.
- If there is a case where aij = bij for i = 1, 2, 3, ...., m and j = 1, 2, 3, ...., n. It indicates that the corresponding elements of both are equal.
In the above image, the first matrix is in the order 2∗2, and the second matrix is in the order 2∗3. So both the matrices do not contain the same number of orders. That's why these matrices are not equal. Suppose we have two more matrices, which are described as follows: In this image, both the matrices contain the order 2∗3. So these matrices will be equal to each other. So we can assign the following values to the matrix: a1 = 1, a2 = 6, a3 = 3, b1 = 5, b2 = 2, b3 = 1. ## Horizontal and Vertical matrixA matrix will be known as the Similarly, a matrix will be known as the The example of calculating the horizontal matrix is described as follows:
## Other types of matricesThere are a lot of other matrices that can be used in discrete mathematics apart from the most commonly used matrix, which we have already explained. All these matrices are described as follows: ## Singular and Non-singular matrixA matrix will be known as a A matrix will be known as a
## Diagonal matrix and Scalar matrixA matrix will be known as the - The diagonal matrix must be a square matrix.
- With the help of form aij, we can characterize the diagonal elements where i = j. It is used to show that a matrix can contain only one diagonal.
Some examples of diagonal matrices are described as follows: In this image, there are three diagonal matrices, P, Q, and R, which contain the order 1∗1, 2∗2, and 3∗3, respectively. A
## Upper and Lower triangular matrixA matrix will be known as the Similarly, a matrix will be known as the
## Symmetric and Skew symmetric matrixIf there is a square matrix D which contains the size n∗n, then this type of matrix will be known as the If there is a square matrix F which contains the size n∗n, then this type of matrix will be known as the
- If there is a square matrix A, in this case, there will be a symmetric matrix A + A
^{T}and a skew-symmetric matrix A - A^{T}. - We can uniquely express the square matrix in the form of addition of a symmetric matrix and a skew-symmetric matrix, which is described as follows:
A = ½(A + A^{T}) + ½(A - A^{T}) = ½(B + C) - Here, B is used to indicate the symmetric matrix, and C is used to indicate the skew-symmetric matrix.
- If there are two symmetric matrices, A and B, in this case, AB will be symmetric AB = BA. That means A and B are commutative.
- If there is a matrix A which is a symmetric or skew-symmetric matrix, in this case, the matrix B
^{T}AB will be symmetric or non-symmetric in correspondence. - In the symmetric matrix, all positive integral powers will always be symmetric.
- If there is a skew-symmetric matrix that contains positive odd integral power, then it will always be skew-symmetric. If there is a skew-symmetric matrix that contains positive even integral power, then it will always be symmetric.
## Hermitian and Skew Hermitian matricesThe hermitian and skew-hermitian matrices are used to have a very small difference, which is described as follows: A given matrix will be known as the A given matrix will be known as the ## Boolean matrixIf all the elements of a given matrix are either 0s or 1s, only then the given matrix will be known as the ## Stochastic matricesIf all the entries of the given matrix represent probability, only then the given matrix will be known as the ## Orthogonal matrixIf there is a square matrix B that contains the relation B∗B Here matrix B is known as the orthogonal matrix because B∗B ## Singleton matrixA matrix will be known as a [], [4], [6], [7], [a], etc ## Special matrixWe have learned various types of matrices, but there are various special kinds of matrices, which are described as follows: ## Idempotent matrixIf there is a square matrix A which contain the relation A ## Nilpotent matrixIf there is a square matrix A that contains the relation A This matrix is a nilpotent matrix because A ## Involutory matrixIf there is a square matrix A that contains the relation A ## Important points- A matrix will be known as the row matrix if there can be any number of columns, but it must contain only 1 row.
- A matrix will be known as the column matrix if there can be any number of rows, but it must contain only 1 column.
- A matrix will be known as a constant matrix if and only if all the elements of a matrix are constant for any given dimension or order of the matrix.
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