All Matrices will have real entries

A matrix is an array of real numbers or entries arranged in a rectangular order consisting of rows and columns, used to represent a mathematical object. These numbers are the elements or entries of the matrix. A matrix with rows titled “p” and columns titled “q” is a matrix of an order of pXq. There are two matrices discussed below:  

Real matrix-matrix whose entries contain real numbers.

Complex matrix-matrix whose entries contain complex numbers.

Types of Matrices that have real entries

Row Matrix- A row matrix is a matrix that has only one row.

Column Matrix- A column matrix is a matrix that has only one column.

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 that has the same number of rows and columns.

Upper triangular Matrix – If all entries of the matrix below the main diagonal are 0, the matrix is an upper triangular matrix. 

Lower triangular Matrix- If all entries of the matrix above the main diagonal are 0, the matrix is a lower triangular matrix.

Diagonal Matrix – If all entries outside the main diagonal are 0, then 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 that is 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 the number of rows or columns is 0.

Singular Matrix- A matrix whose determinant is 0 is known as a singular matrix. 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 . Adjacency matrix is used to denote the pair of vertices that are adjacent or not in a simple graph.

∴ It can be said that all the above types of matrices will have only real entries.

Condition for every matrix

A necessary condition for all the matrices is to have only real numbers and not have any non-real numbers.

The real numbers include rational numbers, natural numbers,  integers, whole numbers, and irrational numbers, i.e. all positive, negative, decimal expansion, algebraic and non-algebraic numbers. 

Non-real numbers are complex numbers that are imaginary in nature, such as roots of the non-positive(negative) integers.

Example- If a matrix has P rows & Q columns, its order can be written as P ✖ Q. If a matrix has order P ✖ Q, then it has PQ elements or entries.

In general, the Apxq matrix has the following array-

A11   A12    A13 – – – – – – – A1Q

A21   A22    A23 – – – – – – – A2Q

AP1   AP2     AP3 – – – – – – APQ

A= [Aij]pxq

1 ≤ i ≤ P, i ≤ j ≤ Q

∴ i,j belongs to N

So, we shall consider only those matrices whose entries are real numbers or functions with real numerical values.

Proof

Suppose matrix A does not have real values.

Let A be a matrix with elements a b c d2x2   

This is a 2 X 2  matrix; where a,b are real numbers with b≠0.panel 

Let λ be an arbitrary value of A.

Matrix A-𝝺I is singular, where I am the 2✖2 identity matrix.

It is equivalent to det(A-𝝺)=0

det(A−𝝺I) = Matrix  =  a-λ   b   -b   a-λ 2×2

 = (a−λ)2−b(−b)

= a2−2aλ+𝝺2+b2

  = 𝝺2−2aλ+a2+b2

 Solving the equation 𝝺2−2aλ+a2+b2 by quadratic formula of D method,

𝝺= 2a±√4a2-4(a2+b2) ∕ 2

= 2a ± √-4b2∕ 2

= a  ±|b|i ; (where b≠0 by assumption)

Hence, λ=a±|b|i isn’t a real number. This holds our assumption untrue.

As λ is an arbitrary element of A, all the elements of A are not real numbers as we have taken b≠0. 

∴ Matrix A will only have real entries.

Computation of the condition

Matrices will only have real entries or elements in their array. Matrices have more general types of entries than real and complex numbers. At the first step of generalisation, any field i.e., a set where addition, subtraction, multiplication, and division operations are well-defined, can be used instead of rational numbers or complex numbers.

For instance- Coding makes the use of matrices over finite fields. Wherever eigenvalues are taken, as these are the roots of a polynomial. They may exist in larger fields than that of the entries of the matrix as they may be complex in the case of a matrix that has real entries.

 Conclusion

In this article, we learned about matrices that only hold real entries or elements, making the matrix with non-real numbers wrong. A matrix is an array of real numbers or entries arranged in a rectangular order consisting of rows and columns, which is used to represent a mathematical object. These numbers are the elements or entries of the matrix. We have also learned the proof of this theorem along with the help of an example. Real matrix-matrix whose entries contain real numbers. Complex matrix- matrix whose entries contain complex numbers.