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Types of Matrices

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Types of Matrices classifies matrices in different categories based on the number of rows and columns present in them, the position of the elements, and also the special properties exhibited by the Matrix. Matrix is a rectangular array of numbers in which elements are arranged in rows and columns. Each element is identified as aij where i and j indicate the row and column number respectively for the element. We have different types of matrices which are classified on the basis of the number of rows and columns, the elements present in them, the order of the matrix, and the properties shown by the matrix. In this article, we will learn about different types of matrices in detail along with a brief introduction to matrices.

What is Matrix?

A rectangular array of numbers, symbols, or any type of entries arranged in the form of rows and columns is called Matrix. A matrix has one or more than one number of rows and columns. The horizontal arrangement of elements is called Row and the vertical arrangement is called the column of matrix. Each element of a matrix is identified by its position which is given by the row and column in which it is present. A matrix is represented by [A]m×n, where m is the number of rows and n is the number of columns present in a matrix. Thus an element of the matrix can be represented as aij where i and j are the ith row and jth column to which an element belongs. Matrix A can be written as:

\begin{bmatrix}  a_{11}& a_{12}  & a_{13}  &.  & . &.  &  a_{1n}\\ a_{21} & a_{22}  & a_{23}  & . &.  &.  & a_{2n} \\ a_{31}& a_{32}  & a_{33}  & . &.  &.  & a_{3n} \\  .&.  &.  &.  &.  &.  &. \\ . &.  &.  &.  &.  &.  &. \\ . &.  &.  &.  &.  &.  &. \\ . & . &.  &.  &.  &.  &. \\  a_{m1}& a_{m2}  & a_{m3}  &.  &.  &.  & a_{mn} \end{bmatrix}_{m\times n}

Example of a Matrix

An example of a Matrix is mentioned below:

\begin{bmatrix}  1& 5  &8   &5 \\ 3& 4  &0   &  12 \\ 7& 2  &3   & 10 \\ \end{bmatrix}_{3\times 4}

Here we can see that we have a matrix of order 3 ⨯ 4 which means there are 3 rows in the matrix and 4 columns in the matrix.

Types of Matrix

There are many types of matrices depending on the elements in the matrix, order, and certain sets of conditions. The different types of matrices are mentioned below:

Let’s learn the above types of matrices in detail

Singleton Matrix

A matrix that has only one element is called a singleton matrix. In this type of matrix number of columns and the number of rows is equal to 1. A singleton matrix is represented as [a]1⨯1.

Example of Singleton Matrix

\begin{bmatrix}  5 \end{bmatrix}_{1\times 1}

In the above example of the Singleton Matrix, there is only one element 5. Hence there is only one column and only one row.

Null Matrix

A matrix whose all elements are zero is called a Null Matrix. A null matrix is also called a Zero Matrix because its all elements are zero. An example of a null matrix is mentioned below:

Example of Zero Matrix

\begin{bmatrix}  0& 0 &0    \\ 0& 0 &0  \\ 0& 0  &0   \\ \end{bmatrix}_{3\times 3}

In the above example of a null or zero matrix, all the elements are zero. Hence the given example is a matrix of order 3 ⨯ 3 whose all elements are zero.

Row Matrix

A matrix that contains only one row and any number of columns is known as a row matrix. A row matrix is represented as [a]1⨯n where 1 is the number of row and n is the number of columns present in a row matrix. An example of a row matrix is given below

Example of Row Matrix

\begin{bmatrix}  1& 3&7 \end{bmatrix}_{1\times 3}

In the above example of a row matrix, the number of rows is 1, and the number of columns is 3. Hence the order of the matrix is 1 ⨯ 3.

Column Matrix

A matrix that contains only one column and any number of rows is called a Column Matrix. A Column Matrix is represented as [a]n⨯1 where n is the number of rows and 1 is the number of columns. An example of a column matrix is given below:

Example of Column Matrix

\begin{bmatrix}  1\\   15 \\ 4\\ 5 \\ \end{bmatrix}_{4\times 1}

In the above example of a column matrix the number of rows is 4 and the number of columns is 1 thus making it a matrix of order 4 ⨯ 1.

Horizontal Matrix

A matrix in which the number of rows is lower than the number of columns is called a Horizontal Matrix. columns

Example of Horizontal Matrix

\begin{bmatrix} 1 & 2 & 3 & 4\\ 5& 6& 7& 8 \end{bmatrix}_{2\times4}

In the above matrix, the number of rows is 2 while the number of columns is 4 thus making it a horizontal matrix.

Vertical Matrix

The matrix in which the number of rows exceeds the number of columns is called a Vertical Matrix. A Vertical matrix is represented as [a]i⨯j where i > j. An example of a Vertical Matrix is mentioned below:

Example of Vertical Matrix

\begin{bmatrix} 1 & 2 \\ 3 & 4\\ 5& 6\\ 7& 8 \end{bmatrix}_{4\times2}

In the above matrix, the number of rows is 4 while the number of columns is 2 thus making it a Vertical matrix.

Rectangular Matrix

A matrix that does not have an equal number of rows and columns is known as a Rectangular Matrix. A rectangular matrix can be represented as [A]m×n where m ≠ n. An example of a rectangular matrix is mentioned below:

Example of Rectangular Matrix

\begin{bmatrix}  1& 3  &7   &15 \\ 3& 4  &6  &  11 \\ 5& 2  &9  & 8 \\ \end{bmatrix}_{3\times 4}

In the above example, we see that the number of rows is 3 while the number of columns is 4 i.e. both are unequal thus making it a rectangular matrix. We can say that both horizontal and vertical matrices are examples of rectangular matrices.

Square Matrix

A matrix that has an equal number of rows and an equal number of columns is called a Square Matrix. Generally, the representation used for the square matrix is [A]n×n. An example of Square Matrix is mentioned below:

Example of Square Matrix

\begin{bmatrix}  8& 3  &2    \\ 6& 4  &6   \\ 5& 7  &9   \\ \end{bmatrix}_{3\times 3}

In the above example of Square Matrix, both the number of rows and columns are 3, thus making them seem like a square structure.

Diagonal Matrix

A matrix that has all elements as 0 except diagonal elements is known as a diagonal matrix. A Diagonal Matrix is only possible in the case of a Square Matrix. An example of a Diagonal Matrix is mentioned below:

Example of Diagonal Matrix

\begin{bmatrix}  8& 0 &0    \\ 0& 4  &0  \\ 0& 0  &9   \\ \end{bmatrix}_{3\times 3}

In the above example, the diagonal elements are 8, 4, and 9 and the rest elements are zero.

Scalar Matrix 

A diagonal matrix whose all diagonal elements are non-zero and the same is called a Scalar Matrix. Scalar Matrix is a kind of diagonal matrix where all diagonal elements are the same. Identity Matrix is a special case of Scalar Matrix.

Example of Scalar Matrix

\begin{bmatrix}  4& 0 &0    \\ 0& 4  &0  \\ 0& 0  &4   \\ \end{bmatrix}_{3\times 3}

In the above example, the given matrix is a diagonal matrix whose all diagonal elements are 4, and hence, this is an example of a Scalar Matrix.

Identity Matrix

A diagonal matrix where all the diagonal elements are 1 and all non-diagonal elements are 0 is called an Identity Matrix. The Identity Matrix is called Unit Matrix. The identity matrix or unit matrix always has an equal number of rows and columns.

Example of Identity Matrix

\begin{bmatrix}  1& 0 &0    \\ 0& 1 &0  \\ 0& 0  &1   \\ \end{bmatrix}_{3\times 3}

In the above diagonal matrix of order 3 ⨯ 3, all the diagonal elements are 1, and non-diagonal elements are zero. Hence this diagonal matrix is an Identity Matrix.

Triangular Matrix

A square matrix in which the non-zero elements form a triangular below and above the diagonal is called a Triangular Matrix. Based on the triangle formed below or above the diagonal, the triangular matrix is classified as:

  • Upper Triangular Matrix
  • Lower Triangular Matrix

Let’s learn them in detail.

Upper Triangular Matrix

A square matrix in which all the elements below the diagonal are zero and the elements from the diagonal and above are non-zero elements is called an Upper Triangular Matrix. In an Upper Triangular Matrix, the non-zero elements form a triangular-like shape.

Example of Upper Triangular Matrix

\begin{bmatrix}  8& 5 &6    \\ 0& 4  &7  \\ 0& 0  &9   \\ \end{bmatrix}_{3\times 3}

In the above example of the Upper Triangular Matrix, all the elements below the diagonal are zero.

Lower Triangular Matrix

A square matrix in which all the elements above the diagonal are zero and the elements from the diagonal and below are non-zero elements is called a Lower Triangular Matrix. In a Lower Triangular Matrix, the non-zero elements form a triangular-like shape from the diagonal and below.

Example of Lower Triangular Matrix

\begin{bmatrix}  8& 0  &0    \\ 6& 4  &0  \\ 5& 7  &9   \\ \end{bmatrix}_{3\times 3}

In the above example of the lower triangular matrix, all the elements above the diagonal are zero.

Singular Matrix

A singular matrix is referred to as a square matrix whose determinant is zero and is not invertible. If det A = 0, a square matrix “A” is said to be singular; otherwise, it is said to be non-singular.

Example of Singular Matrix

A= \left[\begin{array}{ccc} 3 & 6 & 9\\ 6 & 12 & 18\\ 2 & 4 & 6 \end{array}\right]

⇒ |A| = 3(12×6 – 18×4)-6(6×6 – 18×2) + 9(6×4 – 12×2)

⇒ |A| = 3(72 – 72) – 6(36 – 36) + 9(24 – 24)

⇒ |A| = 3×0 – 6×0 + 9×0 = 0

Non Singular Matrix

A Non-Singular matrix is defined as a square matrix whose determinant is not equal to zero and is invertible.

Example of a Non-Singular Matrix

|A| = \left[\begin{array}{cc} 1 & 5\\ 9 & 8 \end{array}\right]

⇒ |A| = 8×1 – 9×5 = 8 – 45 = -37

Symmetric Matrix

A square matrix “A” of any order is defined as a symmetric matrix if the transpose of the matrix is equal to the original matrix itself, i.e., AT = A.

Example of Symmetric Matrix

\left[\begin{array}{cc} 2 & 1\\ 1 & 2 \end{array}\right]

Skew Symmetric Matrix

A square matrix “A” of any order is defined as a skew-symmetric matrix if the transpose of the matrix is equal to the negative of the original matrix itself, i.e., AT = -A.

Example of Skew Symmetric Matrix

\left[\begin{array}{ccc} 0 & 3 & 5\\ -3 & 0 & -2\\ -5 & 2 & 0 \end{array}\right]

Orthogonal Matrix

A square matrix whose transpose is equal to its inverse is called Orthogonal Matrix. In an Orthogonal Matrix if AT = A-1 then AAT = I where I is the Identity Matrix.

Example of Orthogonal Matrix

A = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}

and A^T = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix}

\Rightarrow A \times A^T = \begin{bmatrix} \cos^2(\theta)+\sin^2(\theta) & \cos(\theta)\sin(\theta)-\cos(\theta)\sin(\theta) \\ \sin(\theta)\cos(\theta)-\cos(\theta)\sin(\theta)& \cos^2(\theta)+\sin^2(\theta)\end{bmatrix}

\Rightarrow A \times A^T = \begin{bmatrix}1&0\\0&1 \end{bmatrix} = I_{(2\times 2)}

Idempotent Matrix

An idempotent matrix is a special type of square matrix that remains unchanged when multiplied by itself, i.e., A2 = A.

Example of Idempotent Matrix

A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}

Hence, A \cdot A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = A

Nilpotent Matrix

A Nilpotent is a square matrix that when raised to some positive power results in zero matrix. The least power let’s say ‘p’ for which the matrix yields zero matrix, then it is called the Nilpotent Matrix of power ‘p’.

Example of Nilpotent Matrix

A = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix}

\Rightarrow A^2= \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix}\cdot \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix}

\Rightarrow A^2 = \begin{bmatrix} 0 & 0 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

and A^3 = A \cdot A^2

\Rightarrow A^3 = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Hence, A is a Nilpotent Matrix of index 3.

Periodic Matrix

A periodic matrix is a square matrix which exhibits periodicity, i.e. when raised to some power let’s say p+1 then Ap+1 = A. If p = 1 then A2 = A it means A is an Idempotent Matrix. Thus we can say that Idempotent Matrix is a case of Periodic Matrix.

Example of Periodic Matrix

A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}

The above square matrix is a Periodic Matrix of Period 2, where p = 1.

Involuntary Matrix

An involuntary matrix is a special type of square matrix whose inverse is the original matrix itself, i.e., P = P-1, or, in other words, its square is equal to an identity matrix i.e. P2 = I.

Example of Involuntary Matrix

A = \begin{bmatrix} 2 & 1 \\ -3 & -2 \end{bmatrix}

Hermitian Matrix

A square matrix whose transpose is equal to its conjugate matrix is called Hermitian Matrix.

Example of Hermitian Matrix

A = \begin{bmatrix} 3 + 2i & 1 - i \\ 1 + i & 4 \end{bmatrix}

Skew Hermitian Matrix

A square matrix whose transpose is equal to the negative of its conjugate matrix is called Skew Hermitian Matrix.

Example of Skew Hermitian Matrix

A = \begin{bmatrix} 0 & 2i & -3i \\ -2i & 0 & 4 \\ 3i & -4 & 0 \end{bmatrix}

Boolean Matrix

The matrix which represents the binary relationship and takes 0 and 1 as its element is called Boolean Matrix.

Example of Boolean Matrix

\begin{bmatrix}1 & 0 &1\\0 & 1 & 0\\1&1&0 \end{bmatrix}

Stochastic Matrix

A Square Matrix which represents the probability data i.e. a non-negative element such that the summation of each row is 1.

Example of Stochastic Matrix

\begin{bmatrix}0.2 & 0.5 &0.3\\0.1 & 0.3 & 0.6\\0.4&0.2&0.4 \end{bmatrix}

Trace of a Matrix

The sum of diagonal elements of a matrix is known as the trace of a matrix. The Trace of a matrix A can be represented as tr(A). The Trace of a matrix can be calculated for a square matrix only.

Example:

A= \begin{bmatrix}  15& 12 &9 \\ 4& 6 &11  \\ 5& 9  &0   \\ \end{bmatrix}_{3\times 3}

tr(A) = 15 + 6 + 0 = 21

Properties of a Trace of Matrix

i) Trace of the sum of two matrices is equal to the sum of a trace of an individual matrix.

Explanation:

Mathematically it can be written as tr(A+B) = tr(A) + tr(B)

A= \begin{bmatrix}  15& 12 &9 \\ 4& 6 &11  \\ 5& 9  &0   \\ \end{bmatrix}_{3\times 3}

tr(A) = 15 + 6 + 0 = 21 

B= \begin{bmatrix}  4& 3 &7 \\ 8& 1 &2  \\ 5& 6  &1   \\ \end{bmatrix}_{3\times 3}

tr(B) = 4 + 1 + 1 = 6

Now, tr(A)+tr(B)= 21+6 = 27

A+B= \begin{bmatrix}  19& 15&16 \\ 12& 7 &13 \\ 10& 15  &1   \\ \end{bmatrix}_{3\times 3}

tr(A + B) = 19 + 7 + 1 = 27

You can see, tr(A) + tr(B) = tr(A + tr(B)

Similarly, tr(A – B) = tr(A) – tr(B)

ii) Trace of a matrix that is multiplied by some scalar is equal to the multiplication of the trace of the matrix and scalar.

Explanation:

Mathematically it can be represented as tr(kA) = k tr(A)

A=2\times \begin{bmatrix}  1& 4 &3    \\ 5& 7 &2  \\ 1& 3  &8   \\ \end{bmatrix}_{3\times 3}

tr(2 × A) = 2 + 14 + 16 = 32

2\times A= \begin{bmatrix}  2& 8 &6    \\ 10& 14 &4  \\ 2& 6  &16   \\ \end{bmatrix}_{3\times 3}

2 × tr(A) = 2 * (1 + 7 + 8)

                = 32

You can see tr(2 × A) = 2 × tr(A)

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FAQs on Types of Matrices

Q1: What are Matrices and their Types?

Answer:

Matrices are arrangements of elements or numbers or symbols in the form of an array. The different types of matrices are:

  • Row Matrix
  • Column Matrix
  • Horizontal Matrix
  • Vertical Matrix
  • Singleton Matrix
  • Null Matrix
  • Rectangular Matrix
  • Square Matrix
  • Diagonal Matrix
  • Identity Matrix
  • Scalar Matrix
  • Triangular Matrix
    • Upper Triangular Matrix
    • Lower Triangular Matrix
  • Singular Matrix
  • Non Singular Matrix
  • Symmetric Matrix
  • Skew Symmetric Matrix
  • Orthogonal Matrix
  • Idempotent Matrix
  • Nilpotent Matrix
  • Periodic Matrix
  • Involuntary Matrix
  • Hermitian Matrix
  • Skew Hermitian Matrix
  • Boolean Matrix
  • Stochastic Matrix

Q2: What is a Square Matrix?

Answer:

A Square matrix is a matrix in which the number of rows and columns are equal.

Q3: What is a Diagonal Matrix?

Answer:

A square matrix in which all non-diagonal elements are zero is called a Diagonal Matrix.

Q4: What is Unit Matrix?

Answer:

A diagonal matrix in which all the diagonal elements are 1 is called a Unit Matrix. A unit matrix is also called an Identity Matrix.

Q5: What is a Null Matrix?

Answer:

A matrix whose all elements are zero is called a Null Matrix. A null matrix is also called a zero matrix.

Q6: What is a Symmetric and Skew Symmetric Matrix?

Answer:

A matrix whose transpose is equal to the matrix itself is called a Symmetric Matrix while a matrix whose transpose is equal to the negative of the matrix is called a Skew Symmetric Matrix.



Last Updated : 18 Dec, 2023
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