Types of Matrices

The type of matrix depends on its components, size, and number of rows and columns. This section discusses how different types of matrices are categorized.

A matrix with just one row and any number of columns is a row matrix. Here, the number of columns doesn’t matter, but the matrix will always have just one row. Thus, A = [a_ij]_1n is a row matrix if m = 1. The elements are written horizontally like a list. So, a row matrix is represented as:

A = [a₁₁  a₁₂  a₁₃  …  a_1n]_1n

Example:

A = [1 2] is a 1 × 2 row matrix.

B = [3 2 1] is a 1 × 3 row matrix.

C = [2 3 4 5] is a 1 × 4 row matrix.

Properties of Row Matrix

• A row matrix has a single row.
   
• A row matrix has many columns.
   
• In a row matrix, since all the elements are in a single row, the number of columns equals the number of elements.

A column matrix has only one column and any number of rows. If n = 1, then A = [a_ij]_m1 is a column matrix. The elements are arranged vertically with m rows and 1 column.

  Properties of Column Matrix

• A column matrix has a single column.

• A column matrix has many rows.
   
• Since there is only one column, the number of elements is equal to the number of rows.

A null matrix is a matrix where every element is zero. Therefore, it is also known as a zero matrix. A null matrix is of the order m n, depending on how many rows and columns it has. It is represented as A = [a_ij]_mn, where a_ij = 0 for all i and j.  

Properties Of Zero Matrix (Null Matrix)

• The null matrix can either be square or rectangular, depending on the number of rows and columns.
   
• This null matrix can have an unequal number of rows and columns.
   
• A null matrix is always singular because its determinant is zero (for square matrices), and it doesn’t have an inverse.

A singleton matrix contains only one element. It has one row and one column, so its order is 1 1. 

Properties of Singleton Matrix

• It has only one row and one column.
   
• A singleton matrix contains only one element.
   
• All singleton matrices are square matrices.

A horizontal matrix or row matrix has only one row and any number of columns. Thus, it is represented as [a_ij]_mn. It has the order of

 m × n.

           A = [p q r s]

          Example:

A = [2 2 5 8] is a horizontal matrix of the order 1 × 4.

Properties of Horizontal Matrix

• It has more columns than rows.
   
• It has only a single row.
   
• It is also known as Row Matrix.

It is the matrix in which the number of rows exceeds the number of columns. Thus, it is represented as [a_ij]_mn

Properties of Vertical Matrix

• It has more rows than columns.
   
• If it has only one column, then it is called a column matrix.
   
• It is not square if the number of rows is not equal to the number of columns.

A square matrix has an equal number of rows and columns. Thus, it is represented as [a_ij]_mn. It is of the order m × n, where m = n. 

Properties of Square Matrix

• The number of rows and columns is the same.
   
• Only square matrices have a determinant.
   
• Its transpose is also a square matrix.

A diagonal matrix is a square matrix with zero values for every element outside the main diagonal. It is therefore expressed as A = [a_ij]. Only the main diagonal contains non-zero elements.

Properties of a Diagonal Matrix

• Every diagonal matrix is a square matrix.
   
• The number of rows in a diagonal matrix is equal to the number of columns.
   
• The sum of two diagonal matrices results in another diagonal matrix.

A Scalar matrix is a square matrix in which all the elements of the principal diagonal are the same and all other elements are zero. Therefore, it is written as A = [a_ij]_mn

   Properties of Scalar Matrix

• All the numbers outside the main diagonal are zero.
   
• The determinant of a scalar matrix is equal to the scalar value raised to the power of the order of the matrix. 
   
• It is a special type of square matrix where all the diagonal elements are equal, and all off-diagonal elements are zero.

An identity matrix is a square matrix with ones on the main diagonal and zero elsewhere. Thus, it is represented as A = [a_ij]_mn. The identity matrix works like the number 1 in multiplication, when any square matrix is multiplied by it, the result is the original matrix value.

For any matrix A of order n × n, AI = IA = A

Properties of Identity Matrix

• The identity matrix is always square.
   
• A matrix remains unchanged when it’s multiplied by the identity matrix. 
   
• The determinant is always 1.

When two matrices have the same dimensions, they are considered equal. Thus, it is represented as A = [a_ij]_mn and B = [b_ij]_rs. A and B are equal only if m = r, n = s, and a_ij = b_ij  for all i, j.
         

Properties of Equal Matrix

• Two matrices are equal if they have the same size.
   
• Even if one element is different, the matrices are not equal.
   
• The matrices can be compared for equality only when their orders match exactly.

A triangular matrix is a special type of square matrix. Here, all the elements found above or below the main diagonal are zeros.

Properties of Triangular Matrix

• The inverse of a triangular matrix (if it exists) will also be triangular. 
   
• The determinant of a triangular matrix is found by multiplying all the diagonal values together. 
   
• A triangular matrix can only be inverted if none of the diagonal elements are zero.

A singular matrix is a square matrix whose determinant is equal to zero.

It is represented as |A|=0.

Properties of Singular Matrix

• A matrix is called singular when its determinant turns out to be zero, making it non-invertible.
   
• Since its determinant is zero, a singular matrix doesn’t have an inverse.
   
• A zero matrix is always singular because its determinant is zero.

A matrix is called non-singular if it has an inverse. In other words, it must be square, and its determinant must not be zero.

It can be represented as |A| ≠ 0.

 Properties of Non-singular Matrix

• A non-singular matrix has a non-zero determinant — but not every non-zero matrix is non-singular.
   
• A matrix must have the same number of rows and columns to qualify as non-singular.
   
• A non-singular matrix always has an inverse.

A symmetric matrix is a square matrix where elements are mirrored across the main diagonal — in other words, the entry in row i, column j is equal to the entry in row j, column i.

Properties of Symmetric Matrices

• A symmetric matrix can be diagonalized using an orthogonal transformation.
   
• It has real eigenvalues.
   
• When two symmetric matrices are added or subtracted, the result is also a symmetric matrix.

It is a square matrix where the transpose is equal to the negative of the original matrix.

It is represented as A=[aij].

Properties of Skew-Symmetric Matrix

• In a skew-symmetric matrix, the sum of the diagonal elements is zero.
   
• When a real skew-symmetric matrix is added to the identity matrix, the result will be non-singular.
   
• When a skew-symmetric matrix is squared, the result will be symmetric.

It is a square matrix made up of complex numbers, where the matrix is equal to its own conjugate transpose. 

It is represented as A=A*.

Properties of Hermitian Matrix

• Every Hermitian matrix is a normal matrix.
   
• The sum of two Hermitian matrices is also Hermitian.
   
• If a Hermitian matrix is non-singular, its inverse is also Hermitian.

It is a square matrix with complex entries, where the conjugate transpose is equal to the negative of the original matrix.

It is represented as A* = −A.

Properties of Skew Hermitian Matrix

• A skew Hermitian matrix can be diagonalized.
   
• Its eigenvalues are always purely imaginary or zero.
   
• When a skew Hermitian matrix is multiplied by a scalar, then the result will be skew Hermitian.

An orthogonal matrix is a square matrix with real entries whose rows and columns are orthonormal vectors.
It is represented as AAᵀ =I=AᵀA.

Properties of Orthogonal Matrix

• The inverse of an orthogonal matrix is equal to its transpose.
   
• Multiplying any vector by an orthogonal matrix does not change the vector’s length or the angle between vectors.
   
• Its eigenvalues are either +1 or -1 and eigenvectors are orthogonal.

A square matrix that yields the same matrix when multiplied by itself is called an idempotent matrix.

It is represented as A²=A.

Properties of Idempotent Matrix

• An idempotent matrix is always a square matrix.
   
• It has the same number of rows and columns, making it a square matrix.
   
• If the determinant of an idempotent matrix is zero, it is typically singular.

An involutory matrix is a square matrix that gives the identity matrix when multiplied by itself.
It is represented as A² =I,A^-1=A.

Properties of Involutory Matrix

• When a block diagonal matrix is created using an involutory matrix, the result will also be involutory.
   
• The eigenvalues of an involutory matrix are always either +1 or -1.
   
• The determinant of an involutory matrix is always +1 or -1.

A square matrix that yields a zero matrix for a positive integer power is called a nilpotent matrix.

It is represented as A^p =0.

Properties of Nilpotent Matrix

• The only diagonalizable nilpotent matrix is the zero matrix.
   
• A triangular matrix is said to be nilpotent if its principal diagonal contains all zeros.
   
• A nilpotent matrix is singular, since its determinant is always equal to zero.

Matrices have wide applications in various fields. Here, we will discuss some of those real-life applications.

1. Computer Graphics

Matrices help make simulations, video games, and animations in computer graphics look more realistic and lifelike.

2. Engineering and Physics

Complex problems, such as determining how forces act on a bridge, how electricity flows through circuits, or how heat moves through materials, can be solved with the help of matrices.

3. Cryptography

They contribute to the security of information. When you send secure messages or do online banking, matrices help encrypt data for security reasons.

4. Data Science and Machine Learning

Matrices help process large amounts of data efficiently. They also play a key role in computer decision-making within AI models like neural networks.

5. Computer Vision

Images are stored as number grids, just like matrices. Computers use these grids to sharpen pictures, recognize faces, and even help doctors spot health problems in medical scans.