Introduction to Matrix Class

Eigen Matrix Library

The Eigen Matrix library in C++ programming language is a potent tool for matrix and vector operations, linear algebra operations, geometrical transformations, numerical solving, and other similar algorithms. It is a template library and is defined using header source files. Now, you may ask why we use C++ for performing matrix operations when we have MATLAB and python. C++ is used because - 

• We can implement our algorithms in an embedded system. 
   
• It is an industry-standard language for real-time or embedded systems.
   
• The advantage of Eigen Matrix Library is that we don't have to compile and install anything on our systems; we just need to include the header files.

Eigenvalues

Eigenvalues are a set of scalars related to the linear system of equations in Matrix equations. We can use various arithmetic operations like addition, subtraction, multiplication, etc.

Eigenvectors

An eigenvector is a nonzero vector whose scalar factor can change after applying the linear transformation. It is also called a characteristic vector. It represents directions.  

Installing the Eigen Matrix Library 

To install the Eigen Matrix Library in your system, follow the steps given below:

• Step 1: Click here < select the zip file at the top right corner.
   
• Step 2: Open the zip file in your downloads folder < Eigen < You will find a source folder with all the necessary files. 

Including the Eigen Matrix Library in a Project

Note - We will be working on Microsoft Visuals Studio for this Project. The steps that we are going to follow can be generalised to any compiler that you might prefer.

Step 1: Install Eigen matrix toolbox < Add the Eigen Matrix toolbox to your Project.
 

Step 2: Open the zip file < copy the source folder < and paste it into a new folder.
 

Step 3: Open a new Microsoft visual project < Windows Console Application < Okay
 

Step 4: When a new project is created, Right-click on console application < Properties < C++ < Precompiled headers < Select Not-using precompiled headers < Okay

Step 5: Adding Eigen source files to our project path.

• Right-click on console application < Properties < C++ < General < Additional Include directories < Select the folder wherein we pasted the source folder < Okay

Voila! Our source and header files are added to our project path.

Matrix Operations

This section of the Introduction to Matrix class covers the various Matrix Operations that the Eigen Matrix Library has to offer. Listed below are the various matrix operations we are going to discuss.

Addition and Subtraction

Things to keep in mind while performing addition and subtraction operations -

• Both sides, i.e., left and right must have an equal number of rows and columns.
   
• They must be of the same scalar type.
   

We will be discussing the following three operators here -

• Binary operators (a+b) and (a-b)
   
• Unary operator (-a)
   
• Compound operator (a+=b) or (a-=b)
   

Let us understand the addition and subtraction matrix operation through an example -

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{  
   // Define two matrices A and B (2X2)
   Eigen::Matrix2d A(2,2);
   A << 5,6,7,8;
   Eigen::Matrix2d B(2,2);
   B << 1,2,3,4;
   
   // Performing addition, subtraction and compound operator on them.
   cout << "A + B =" <<endl<< A+B << endl;
   cout << "A - B = " <<endl<<A-B << endl;
   cout << "A +=B;" << endl;
   A += B;
   cout << "New A =" <<endl<< A << endl;

}

Input

A =

B =

Output

Scalar Multiplication and Scalar Division

In this section of our article, Introduction to Matrix class, we will learn about the multiplication and division of scalars. The matrix operations that we can perform are as follows - 

• Binary operator - ( * ) (matrix*scalar & scalar*matrix)
   
• Binary operator - ( / ) (matrix/scalar)
   
• Compound operator - ( *= ) (matrix*=scalar)
   
• Compound Operator - ( /= ) (matrix/=scalar)
   

Let us understand the scalar multiplication and scalar division matrix operation through an example -

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
   // Define and initialise a matrix m
   Eigen::Matrix2d m(2,2);
   m << 2,4,6,7;
   
   // Matrix multiplication with a scalar
   cout << "m * 2 =" <<endl<< m * 2 << endl;
  
   // Matrix division with a scalar
   cout << "m / 2 =" << endl << m / 2 << endl;
   
   // Using compound operators
   m *= 2;
   cout << "m *= 2 :" << endl << m <<endl;
   m /= 4;
   cout << "m /= 4 :" << endl << m <<endl;
 
}

Input

M =

Output

Transposition and Conjugation

There are predefined matrix operations available to determine a matrix's transpose, conjugate, and adjoint. Let us first understand what these terms mean.

Transpose

If, in a matrix, we switch the elements of the rows with the elements of the columns, we obtain the transpose of a matrix. For example -

Conjugate

When we replace all matrix elements with their respective complex conjugate numbers, the resulting matrix is conjugate. For example - 

Adjoint

The transpose of a cofactor matrix is the adjoint of the matrix. For example -

Let us understand the transpose, conjugate, and adjoint matrix operation through an example -

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;

int main()
{
   //Creating a 2X2 matrix using eigen
   Eigen::Matrix2d A(2,2);
   A << 10, 20, 30, 40;
   cout << "Matrix A" <<endl<< A << endl;
  
    //Transpose of matrix A
   cout << "Transpose of A" <<endl<< A.transpose() << endl;
  
    //Conjugate of matrix A
   cout << "Conjugate of A" <<endl<< A.conjugate() << endl;
  
    //Adjoint of matrix A
   cout << "Adjoint of A " <<endl<< A.adjoint() << endl;
}

Input

A = 

Output

Matrix-matrix and Matrix-vector Multiplication

The matrix operators used for performing matrix-matrix multiplication are -

• Binary operator - ‘ * ‘ (matrix*scalar & scalar*matrix
   
• Compound operator - ‘ *= ‘ (matrix*=scalar)

Let us understand the matrix-matrix and matrix-vector multiplication matrix operation through an example -

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
    Eigen::Matrix2d A;
    A << 1, 2, 3, 4;
    Eigen::Vector2d u(-1, 1), v(2, 0);
    
    cout << "Here is A*A:\n" << A * A <<endl;
    cout << "Here is A*u:\n" << A * u << endl;
    cout << "Here is u^T*A:\n" << u.transpose() * A <<endl;
    cout << "Here is u^T*v:\n" << u.transpose() * v << endl;
    cout << "Here is u*v^T:\n" << u * v.transpose() << endl;
    cout << "Multiplying A with itself" << endl;
    
    A = A * A;
    cout << "Now A is A:\n" << A<< endl;
}

Input

Matrix =

Output

Dot product and Cross product

In this section of our article, Introduction to Matrix class, we will discuss the dot and cross-product operations. To perform dot and cross-product operations, we use dot() and cross() functions. The use of these functions is depicted in the following example - 

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
   Eigen::Vector3d v(10, 20, 30);
   Eigen::Vector3d w(0, 1, 2);

   cout << "Dot product:" << v.dot(w) << endl;
  
   float dp = v.adjoint() * w;
   
   cout << "Dot product via a matrix product: " << dp << endl;
   cout << "Cross product:" <<endl<<v.cross(w) << endl;
}

Input

v={10,20,30}
w={0,1,2}

Output

Basic Arithmetics Reduction Operations

Apart from the operations discussed in the adobe sections, we also have some predefined functions that come with the Eigen Matrix Library. Let us understand why they are implemented and how we can use them with the help of an example - 

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
   Eigen::Matrix2d A;
   A << 1, 2,3, 4;
   
   cout << "Matrix Sum is : " << A.sum() << endl;
   cout << "Matrix Product is : " << A.prod() << endl;
   cout << "Matrix Mean is :  " << A.mean() << endl;
   cout << "Matrix MinCoefficient is :  " << A.minCoeff() << endl;
   cout << "Matrix MaxCoefficient is : " << A.maxCoeff() << endl;
   cout << "Matrix Trace is : " << A.trace() << endl;
}

Input

Matrix A= 

Output

Now that we have covered the matrix operators section of our blog, Introduction to matrix class, let’s move on to Eigen’s array class

Eigen’s Array Class

Before getting on to the implementation and use of Eigen's array class, let us understand what an Eigen array class is and the different Eigen array class types.

What is an Eigenarray class?

In this section of our article, Introduction to Matrix class we will discuss the Eigenarray class. Both the Eigen array class and Eigen matrix class are somewhat similar. However, one of the differences between them is that Eigen's array class provides the users with general-purpose arrays, and its API makes it easier for them to access coefficient-wise operations. Whereas Eigen's matrix class is purely for linear algebra, its API makes it easier to access operations related to linear algebra. 

Eigen Array Class Type

Array refers to both 1D and 2D arrays in this section. The ArrayNt form of typedefs represents 1D arrays, and the ArrayNNt form represents 2D arrays. Here, N is the size type, and t is the scalar type.

Accessing Values inside an Array

The following example shows how we can assign values in an array - 

Implementation

#include <Eigen/Dense>
#include <iostream>
using namespace std;
int main()
{
   Eigen::ArrayXXf  A(2, 2);

   // Initialise matrix coefficient by coefficient
   A(0, 0) = 1.0; A(0, 1) = 2.0;
   A(1, 0) = 3.0; A(1, 1) = A(0, 1) + A(1, 0);

   // Output values
   cout << A << endl ;

   // Using the comma-initializer is valid syntax
   A << 1.0, 2.0,
       3.0, 4.0;

   // Output values
    cout << A << endl;
}

Output

Addition and Subtraction in Arrays

Note - While performing arithmetic operations in an array, the size of the arrays must be the same. Let's take a look at a code to understand its implementation better - 

Implementation

#include <Eigen/Dense>
#include <iostream>
using namespace std;

int main()
{
   Eigen::ArrayXXf M1(3, 3);
   Eigen::ArrayXXf M2(3, 3);
   
   M1 << 1, 2, 3,
       4, 5, 6,
       7, 8, 9;
   M2 << 1, 2, 3,
       1, 2, 3,
       1, 2, 3;

   // Adding the two arrays
   std::cout << "M1 + M2 = " <<endl << M1 + M2 << endl << endl;

   // Subtracting a scalar from an array
   std::cout << "M1 - 2 = " << endl << M1 - 2 <<endl;
}

Input 

M1 = 

M2 = 

Output

Array Multiplication

In this section of our article, Introduction to Matrix class, we will discuss array multiplication. Array multiplication can only be performed if their dimensions are similar to the product obtained by the multiplication of arrays and are interpreted coefficient-wise. 

Implementation

#include <iostream>
#include <Eigen/Dense>
using namespace std;
int main()
{
   Eigen::ArrayXXf M1(2, 2);
   Eigen::ArrayXXf M2(2, 2);
   
   // Initialising two arrays[2X2]
   M1 << 1, 2,
       3, 4;
   M2 << 5, 6,
       7, 8;
  
   // Multiplying two arrays
   cout << "M1 * M2 = " << endl << M1 * M2<< endl;
}

Input

M1 = 

M2 = 

Output

Other Coefficient-wise Operations

Apart from the basic operations discussed in the above sections, Eigen's array class also offers various coefficient-wise operations. For example, the .sqrt() method lets the user determine the square root of any given number.

Implementation

#include <Eigen/Dense>
#include <iostream>
using namespace std;
int main()
{
   Eigen::ArrayXf M1 = Eigen::ArrayXf::Random(5);
   M1 *= 2;
   
   cout << "M1 = " <<endl
       << M1 << endl;
   cout << "M1.abs() = " <<endl
       << M1.abs() << endl;
   cout << "M1.abs().sqrt() =" << endl
       << M1.abs().sqrt() << endl;
   cout << "M1.min(M1.abs().sqrt()) = " << endl
       << M1.min(M1.abs().sqrt()) <<endl;
}

Input

Here, M1 is an array of five random numbers.

Output

 

Check out this problem - Matrix Median

Frequently Asked Questions

What is a Matrix?

A matrix is an arrangement of numbers or functions arranged in a rectangular array consisting of several rows and columns. For example - a matrix with A number of rows and B number of columns is represented as A x B.

What are Eigenvalues?

Eigenvalues are a set of scalars related to the linear system of equations in Matrix equations. We can use various arithmetic operations like addition, subtraction, multiplication, etc.

What are Eigenvectors?

An eigenvector is a nonzero vector whose scalar factor can change after applying the linear transformation. It is also called a characteristic vector. It represents directions.  

How can we find the adjoint of a matrix?

To find the adjoint of a matrix, we first see the cofactor and then transpose it.

What is the conjugate of a matrix?

If all matrix elements are replaced by their respective conjugates, we get the conjugate of that matrix.

Conclusion

This article, Introduction to Matrix class, explored the Matrix class in Eigen, different types of matrix operations, and how to use Eigen's array class. If you want to dig deeper into Eigen, here are some related articles - 

• Getting started with Eigen
   
• How to use Eigen's array class
   
• What happens inside Eigen?

 

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