Introduction

In previous entries we saw how to define matrices in programming and load data into them, in this article we will see how to multiply two matrices using two-dimensional arrays.

Initial data

We will start from two matrices “A” and “B” with preloaded data and we will analyze a method that will multiply two matrices that are sent as parameter and returns the result matrix.

In figure 1 we see the declaration of two 2×2 matrices and the loading of their data, then in lines 27 and 28 we make the calls to the method that is in charge of multiplying the matrices that are sent as parameters and returning the resulting matrix.

Algorithm for multiplying matrices in programming

The product of matrices is not commutative, so it matters the order in which the matrices are multiplied. Let’s consider that matrix A is on the left and multiply it by matrix B which is on the right.

The procedure to multiply two matrices consists of multiplying each element of the i-th row of the left matrix by each element of the j-th column of the right matrix and then adding these products, the result of that sum will be the ij-element of the resulting matrix. Therefore, in order to carry out the product of matrices, the condition that the number of columns of the left matrix equals to the number of rows of the right matrix must be fulfilled.

The result of the multiplication of matrices A and B will be another matrix that will have a number of rows equal to the number of rows in A and a number of columns equal to the number of columns in B.

Algorithm analysis

Figure 2 shows the implementation of an algorithm that multiplies two matrices that are sent as parameters. First the result matrix is declared and the sizes of both matrices are read, this is done in lines 32 to 36.

Then we check if it is possible to multiply both matrices, if it is not possible the result will be null, otherwise we proceed to solve the product of matrices.

The resulting matrix structure is created with the corresponding number of rows and columns (see the description of the procedure above). Then the three nested for loops shown in figure 2 are used to determine the value of each resulting element.

Introduction

In previous entries we saw how to define matrices in programming and load data into them, in this article we will see how to add and subtract two matrices using two-dimensional arrays.

To be able to add or subtract two matrices it is necessary that both have the same size, that is to say the same number of rows and the same number of columns, otherwise the operation cannot be solved. We will take this into account in our algorithm, at the time of performing the operation we will check that these conditions are met and if everything is correct the sum will be solved, otherwise the returned matrix will have a null value and this result can be used to make subsequent checks, for example if the resulting matrix is different from null we consider that the operation was successful.

Initial data

We will start from two matrices “A” and “B” with preloaded data and we will analyze a method that will perform the sum of two matrices that are sent as parameter and returns the result matrix.

In figure 1 we see the declaration of two 2×2 matrices and the loading of their data, then in lines 28, 29 and 30 we make the calls to the methods that will be in charge of adding, subtracting or making a linear combination between the matrices that are sent as parameter, those methods will return the resulting matrix.

Algorithm to add two arrays in programming, C# language

To make the sum of two matrices we must first make sure that both matrices have the same size, if this is correct we proceed to go through each element of the matrices and perform the sum element by element, assigning the result to a new matrix previously declared.

Analysis of the algorithm that performs the addition of two matrices

Between lines 30 and 56 we have defined a method that receives as parameters two two-dimensional arrays (“matA” and “matB”) and returns another two-dimensional array.

The result matrix is declared on line 32 and returned on line 55, at the end of the procedure.

Then in line 34 we have an IF statement to check if it is possible to solve the sum of the matrices, the condition is that the matrices match in number of rows and columns, if this is true we proceed to solve the sum, otherwise a message is printed on the console indicating that it is not possible to solve the operation and the result variable is assigned null (lines 51 and 52).

If the sum of matrices can be solved we will determine the number of rows and columns of the result matrix and we will create the data structure for that matrix, we do this in lines 36, 37 and 39 respectively.

The addition of the matrices is done element by element, therefore we will use two nested loops, the first one will go through each row and the inner loop will go through each column, so we will start with a row and go through each of the columns, at the end we will go to the next row. Inside the inner loop we simply solve the sum of the ij-elements of each matrix and assign it to the ij-element of the result matrix.

Algorithm to subtract two matrices in programming, C# language

To subtract two matrices in programming, the algorithm is practically the same as the addition, only that when their elements are traversed, instead of adding them, they are subtracted, we can see an implementation of the algorithm in Figure 3.

Algorithm to perform the linear combination of two matrices in programming, C# language.

Since the algorithms for adding and subtracting matrices are very similar, a function could be defined that performs a linear combination between the two matrices indicated together with their coefficients, i.e. it solves the operation:

MatrixC = a * MatrixA + b * MatrixB

In this way to perform the subtraction of matrix A minus matrix B we could call this function passing as parameter 1 for the coefficient that multiplies matrix A and -1 for the coefficient that multiplies matrix B, but we also have a function with higher capabilities. In Figure 4 we see the algorithm that solves the linear combination between both matrices.

Introduction

In previous entries we saw how to define matrices in programming and load data into them, in this article we will see an algorithm to transpose a matrix in programming, using C# language in Unity.

Initial data

We will start from an “A” matrix with preloaded data and we will analyze a method that will perform the transposition of the matrix sent as parameter and will return the result matrix.

Algorithm for transposing matrix of N rows and M columns

Transpose a matrix implies transforming its rows into columns and its columns into rows, therefore the resulting matrix will be a matrix that will have its dimensions exchanged. The function that will be in charge of transposing the matrix that enters as parameter is shown in figure 2, first we will read the number of rows and columns of the input matrix and we will create the structure of the result matrix, this is done in lines 30, 31 and 32 respectively, notice how in the creation of the result matrix, the dimension is exchanged comparing with the input matrix.

Then we have to traverse each element of the input matrix, we have seen that this could be done by two nested for-loops, the first one traverses rows and the inner loop traverses columns. Inside the loop what we will do is to assign the ij-element of the input matrix, to the ji-element of the output matrix.

At the end of the loop, the result matrix is returned (line 41).

Introduction

In mathematics, a matrix is a structure containing numerical data arranged in rows and columns, one of its main uses is to represent and solve systems of linear equation.

In this article you will see how to represent a matrix and load data to it in C# language in Unity Engine.

About the development environment, script creation and execution

For this example we will use the Unity engine, create a C# Script and assign it to a GameObject in the scene, when pressing the Play button, Unity will automatically execute some functions inside the Script.

Implementation of a matrix in programming

Declaration of data

Matrices have a certain number of rows and columns, so we will need two integers, let’s call them “rows” and “columns”.

To declare the matrix in programming we will use a two-dimensional arrays and we must indicate the data type, usually we are going to work with the set of real numbers, so the data type we are going to use is “float”. However we could define arrays of any data type needed, for example “bool”, “int”, “double”, “string”, etc.

The declaration of the required data would be as follows:

To declare a two-dimensional array, brackets are used and a comma is placed for each extra dimension that is needed, in this case as we need two dimensions it looks like this: “[ , ]”.

Initialization of the matrix data

In the previous step we declared the necessary variables, now it is time to initialize those variables, that is to say to give them concrete values or to create data structures and assign them. The initialization will be done in the “Start” method of the Script, the Start and Update functions are automatically defined when creating a new Script. By pressing the Play button in Unity, the Start functions of all the scripts present in the scene will be automatically executed, this will happen before the the first frame of the program appears on the screen.

In figure 2 we see an example of the initialization of a 2×2 matrix, in lines 17 and 18 is the initialization of the rows and columns respectively. In line 20 a two-dimensional array is created indicating the total number of elements of each dimension and this array is assigned to the “myMatrix” field that was declared in the previous step.

In C# the first element of the arrays is 0 and the last is “n-1”, where n is the size of the array, so in line 22 for example, the data in position [0,0] refers to the element in the first row and first column.

Conclusion

We have seen how to declare two-dimensional arrays that can be used to represent matrices and subsequently perform operations between them.

In this example the data type used for the array is “float”, which means that each element will have a number with decimal point stored, but you could define arrays of any data type you need, for example “bool”, “int”, “double”, “string”.

To read or modify an element within the matrix we use the name with which it was defined and between brackets we indicate the row and column separated by comma, so for example if we write “myMatrix[1,0]” we are pointing to the element of the matrix that is in the second row, first column.