C++ Program to Check if a Matrix is Orthogonal or NotIn this article, we will discuss a C++ program to check if a matrix is orthogonal or Not with its output. But before going to the program, we must know about the orthogonal. The orthogonal matrix is one in which the transpose of the original matrix and its inverse matrix are identical. If we have a square matrix, we may consider it an orthogonal matrix if its transpose and inversion are comparable to each other, i.e., AT = A-1. In this case, AT represents the transpose of matrix A, and A-1 represents the inverse of matrix A. On the basis of this definition, there is one different definition of an orthogonal matrices, which will be explained as follows: AT = A-1
Steps to check whether the given matrix is orthogonal or not:Various steps must be taken to find out whether the matrix is orthogonal or not. We'll use a square matrix A for this. The steps involved are as follows: Step 1: In this step, we are going to find the determinant of the provided matrix. If the determinate of matrix A is 1, the matrix is orthogonal. Step 2: In this step, we will find the transformed version of this matrix and its inverse. Step 3: The matrix is said to be orthogonal if the product of the transpose of matrix A and the inverse of matrices A is an identity matrix, i.e., AT*A-1 = I. It won't be an orthogonal matrix anymore. I denote the identity matrix. Determinant of an Orthogonal matrix:If we compute the determinant of orthogonal matrices, it remains either +1 or -1. Now, we will explain this. We'll use an orthogonal matrix A for this. We are able to determine from its definition that. A*AT = I Now, we will compute the determinant on the two sides of the above equation and obtain the following results: det(I) = det(A*AT) We understood that if we calculate the determinant of a matrix of identity, we get 1. As well, if A and B are matrices, det(AB) = detA*detB. So, AAT= AA-1 As we have seen, AA-1 = I, where I is an identity matrix, which has the same rank as matrix A. As a result, AA-1= I. Similarly, we can show that AT A = I. We can get the following expression from the previous two: ATA = AAT = I. Example:Let us take a C++ program to check for a matrix is Orthogonal or not. Approach 1: Output: The given matrix is an Orthogonal Matrix Approach 2: Combining three traversals is an optimal solution. Instead of finding transposition explicitly, we make use of array[j][k] instead of array[k][j]. Also, rather than calculating the product specifically, we check identity while computing the product. Example: Let us take another C++ program to check for a matrix is Orthogonal or not. Output: The given matrix is an Orthogonal Matrix Orthogonal matrix applications:The orthogonal matrix has several applications or uses, some of which are listed below:
|