Matrix and Matrix Types

1. Matrix and Matrix Types
A rectangular array of m × n numbers (real or complex) in the form of m horizontal lines (called rows) and n vertical lines (called
columns), is called a matrix of order m by n, written as m × n matrix. Such an array is enclosed by [ ] or ( ). In this article, we will learn
the meaning of matrices, types of matrices, important formulas, etc. There are different types of matrices that are categorized based
on the value of their elements, their order, the number of rows and columns, etc. Now, using different conditions, the various matrix
types are categorized below, along with their definition and examples.
• Zero Matrix: If all the elements are zero in a matrix, then it is called a zero
matrix, generally denoted by 0. Thus, A = [aij]mxn is a zero-matrix if aij = 0 for
all i and j.
• Square Matrix: A square matrix is a matrix where the number of rows (n)
equals the number of columns (m). The square matrix is contrasted with
the rectangular matrix where the number of rows and columns are not
equal. Given that the number of rows and columns match, the dimensions
are usually denoted as n, e.g. n x n. The size of the matrix is called the
order, so the order 4 square matrix is 4 x 4. The vector of values along the
diagonal of the matrix from the top left to the bottom right is called the main
diagonal.

2. • Triangular Matrix: A triangular matrix is a type of square matrix that has all
values in the upper-right or lower-left of the matrix with the remaining
elements filled with zero values.
1. Upper Triangular Matrix: A triangular matrix with values only above the
main diagonal is called an upper triangular matrix.
2. Lower Triangular Matrix: A triangular matrix with values only below the
main diagonal is called a lower triangular matrix.
• Diagonal Matrix: A diagonal matrix is one where values outside of the
main diagonal have a zero value, where the main diagonal is taken from
the top left of the matrix to the bottom right. A diagonal matrix is often
denoted with the variable D and may be represented as a full matrix or as a
vector of values on the main diagonal.
• Identity Matrix: An identity matrix is a square matrix that does not change
a vector when multiplied. The values of an identity matrix are known. All of
the scalar values along the main diagonal (top-left to bottom-right) have the
value one, while all other values are zero. An identity matrix is often
represented using the notation “I” or with the dimensionality “In”, where n is
a subscript that indicates the dimensionality of the square identity matrix. In
some notations, the identity may be referred to as the unit matrix, or “U”.
• Scalar Matrix: If all the elements in the diagonal of a diagonal matrix are
equal, it is called a scalar matrix.