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A matrix is a rectangular array of a set of a number. It is a 2-dimensional rectangular array. It has rows that are represented by ‘m’ and columns which are represented by ‘n’. The dimension of a matrix is represented by x n’, which represents the number of rows and columns, respectively.
Row Matrix: A row matrix is a matrix that has only one row of elements.
Column Matrix: A column matrix is a matrix that has only one column of elements.
Singleton Matrix: If there is only one element in the matrix it is called a singleton matrix.
Horizontal Matrix: A matrix with order P x Q is a horizontal matrix if Q > P.
Vertical Matrix: A matrix with order P x Q is a vertical matrix if P > Q.
Equal Matrices: Equal matrices are those matrices that have an equal number of elements.
Square Matrix: A square matrix is a matrix with the same number of rows and columns.
Upper Triangular Matrix: If all matrix entries below the main diagonal are 0, the matrix is an upper triangular matrix.
Lower Triangular Matrix: If all matrix entries above the main diagonal are 0, the matrix is lower triangular.
Diagonal Matrix: If all entries outside the main diagonal are 0, it is a diagonal matrix.
Identity Matrix: The identity matrix In of size n is the n-by-n matrix in which all the entries on the main diagonal are 1, and all other entries are 0.
Symmetric or skew-symmetric Matrix: A square matrix equal to its transpose is a symmetric matrix.
Invertible Matrix: A square matrix is called invertible if there exists a matrix B such that
AB = BA = In .
Definite Matrix: A symmetric matrix is positive-definite if its eigenvalues are positive, i.e. the matrix is positive, semi-definite, and invertible.
Orthogonal Matrix: An orthogonal matrix is a matrix where rows & columns are orthogonal unit vectors, & its transpose is equal to its inverse.
Empty Matrix: An empty matrix is a matrix in which rows or columns are 0.
Singular Matrix: A matrix whose determinant is 0 is a singular matrix. The singular matrix has no inverse.
Adjacency matrix or Connection matrix: A matrix in which graph vertices denote the rows and columns with 0 and 1. The Adjacency matrix denotes the pair of adjacent or not vertices in a simple graph.
Symmetric matrix: A symmetric matrix is defined as the matrix, which is also a square matrix and is equal to its transpose. The column n and row m values are the same in the square matrix, i.e. m=n.
Example Suppose a 3 x 3 matrix represented by S=
And its transpose is represented by ST
=
Hence, we can see S=ST.So S is a symmetric matrix.
So S is a symmetric matrix that lets us understand some of the properties of a symmetric matrix
1. A symmetric matrix should be a square matrix in which the number of rows and columns must be equal
2. Suppose a matrix is represented by F, then F. Ft is symmetric
3. Any multiple of a symmetric matrix of F is also symmetric.
4. The transpose of the matrix should be equal to its original matrix
5. Suppose there are two matrices, F and H, then
FH is considered symmetric when FH=HF, which is the commutative property.
Addition and subtraction between the F and H matrix is also symmetric.
6. Suppose the F matrix is symmetric. Fn is also symmetric up to an nth term of F.
7. The inverse of the F matrix is symmetric only when F is symmetric.
Skew symmetric matrix
A square matrix that is skew-symmetric and attributes is always known as a non-symmetric matrix. In the square matrix, the column n and row m values are the same, i.e. m=n, but the transpose of a matrix is negative.
Suppose we have a 3 X3 square matrix S, then
ST=-S
Example:- S =
ST=
Here we can see ST=-S. So S is a skew-symmetric matrix.
Some of the theorems associated with Symmetric matrices include the following:
1. A square matrix can be written as the sum of skew-symmetric matrices and symmetric matrices.
Suppose a matrix is defined as F then in its Symmetric matrix, Ft=F
2. Skew symmetric matrix is represented by Ft = -F
Here we can denote a square matrix J as the combination of skew-symmetric matrix and symmetric matrix As, J=(F-Ft)/2 + (F+Ft)/2
3. In General, Square matrices are the products of a symmetric matrix.
Conclusion
This article has studied the matrices, type of matrices, symmetric and skew-symmetric matrices, and their properties. Asymmetric matrix is defined as the matrix that is also a square matrix and is equal to its transpose. Here the square matrix is the matrix whose number of rows m and columns n should be the same, i.e. m=n. A non-symmetric non-symmetric matrix is also known as a skew-symmetric matrix. In this matrix type, the matrix is square, i.e. the number of rows m and columns n should be the same, i.e. m=n, but the transpose of a matrix is negative. And the theorems, which include A square matrix, can be written as the sum of skew-symmetric matrices and symmetric matrices, and Square matrices are the products of a symmetric matrix.
