Q: What is the algorithm to generate transpose of given matrix?

Write your answer...

Submit

Still have questions?

Continue Learning about Math & Arithmetic

The matrices that follow d rule of reflexivity is known as ref matrix

In computer science, deterministic algorithm is an algorithm which, given a particular input, always produces the same result. This is used to increase the efficiency of machines.

A matrix is symmetric when it is a perfect square and the transpose of the matrix is equal to the original matrix. The implementation is reasonably straightforward and can be done without actually creating the transposed matrix. We simply imagine a dividing line through the main diagonal (top-left to bottom right) and compare the corresponding elements on each side of that diagonal.bool is_symmetric (int* a, int rows, int cols) {if (rows!=cols) return false;for (int row=0, row

#include<stdio.h> #include<math.h> float detrm(float[][],float); void cofact(float[][],float); void trans(float[][],float[][],float); main() { float a[25][25],k,d; int i,j; printf("ENTER THE ORDER OF THE MATRIX:\n"); scanf("%f",&k); printf("ENTER THE ELEMENTS OF THE MATRIX:\n"); for(i=0;i<k;i++) { for(j=0;j<k;j++) { scanf("%f",&a[i][j]); } } d=detrm(a,k); printf("THE DETERMINANT IS=%f",d); if(d==0) printf("\nMATRIX IS NOT INVERSIBLE\n"); else cofact(a,k); } /******************FUNCTION TO FIND THE DETERMINANT OF THE MATRIX************************/ float detrm(float a[25][25],float k) { float s=1,det=0,b[25][25]; int i,j,m,n,c; if(k==1) { return(a[0][0]); } else { det=0; for(c=0;c<k;c++) { m=0; n=0; for(i=0;i<k;i++) { for(j=0;j<k;j++) { b[i][j]=0; if(i!=0&&j!=c) { b[m][n]=a[i][j]; if(n<(k-2)) n++; else { n=0; m++; } } } } det=det+s*(a[0][c]*detrm(b,k-1)); s=-1*s; } } return(det); } /*******************FUNCTION TO FIND COFACTOR*********************************/ void cofact(float num[25][25],float f) { float b[25][25],fac[25][25]; int p,q,m,n,i,j; for(q=0;q<f;q++) { for(p=0;p<f;p++) { m=0; n=0; for(i=0;i<f;i++) { for(j=0;j<f;j++) { b[i][j]=0; if(i!=q&&j!=p) { b[m][n]=num[i][j]; if(n<(f-2)) n++; else { n=0; m++; } } } } fac[q][p]=pow(-1,q+p)*detrm(b,f-1); } } trans(num,fac,f); } /*************FUNCTION TO FIND TRANSPOSE AND INVERSE OF A MATRIX**************************/ void trans(float num[25][25],float fac[25][25],float r) { int i,j; float b[25][25],inv[25][25],d; for(i=0;i<r;i++) { for(j=0;j<r;j++) { b[i][j]=fac[j][i]; } } d=detrm(num,r); inv[i][j]=0; for(i=0;i<r;i++) { for(j=0;j<r;j++) { inv[i][j]=b[i][j]/d; } } printf("\nTHE INVERSE OF THE MATRIX:\n"); for(i=0;i<r;i++) { for(j=0;j<r;j++) { printf("\t%f",inv[i][j]); } printf("\n"); } } ALTERNATIVE CODE: #include<iostream.h> #include<stdio.h> #include<conio.h> #include<process.h> #include<math.h> // Written by Ran // There have been enough comments to help the reader easily understand this program // Helpfulness of COMMENTS in this program and Pre-requisites:- // a. However it's assumed that the reader is familiar with the basics of C++ // b. It is also assumed that the reader knows the basic mathematics involving matrices. // c. Since this program focusses on how to find inverse of a matrix, the comments // in the program are sufficient for understanding this. // It is assumed that the reader is familiar with basics of matrices in C++ (like input, display, // addition,transpose,etc. of matrices) // The comments in this program aim to explain the reader how to find inverse // d. Hence through comments, the reader will also be taught how to find determinant, // adjoint, cofactor,etc. However as said in the previous lines, there won't be comments // for explaining basics like input,display,etc of a matrix using C++. // NOTES: // 1. float datatype has been used for matrix, determinant. // 2. To have consistency between mathematics and C++, this program considers a[1][1] as the first element // i.e row and column indices begin with 1 same as mathematics. // Define a structure matrix with a matrix (2D array of type float) and size n // Declare the objects of this structure used in this program struct matrix { float a[25][25]; int n; }obj,c_obj,trans_obj,obj_cof,obj_adj,obj_inv; // Prototypes of the functions used in this program void input(matrix&); void display(matrix&); matrix reduced(matrix &, int ,int ); float determinant(matrix); float cofactor(matrix,int,int); matrix transpose(matrix); matrix adjoint(matrix); matrix inverse(matrix obj); // Begining of Main function int main() { // Getting dimensions input by the user int r,c; again: cout<<"Enter the order of the matrix: "<<endl; cout<<"Enter Row dimension: "; cin>>r; cout<<"Enter Column dimension: "; cin>>c; // Check dimensions for square matrix so that inverse can be found // If user enters different dimensions for row and column, ask to re-enter or quit program if(r!=c) { char ans; cout<<"Inverse can be found out only for a square matrix. Enter same dimension for row and column. Do you want to enter the dimensions again? Press Y for yes"<<endl; cin>>ans; if(ans=='y') goto again; else cout<<"Program exit"; getch(); exit(0); } // If it's a square matrix, proceed else if(r==c) {obj.n=r;} cout<<endl; input(obj); // call input function to input the matrix elements from the user display(obj); // display the matrix got as input now // Following lines were used to test parts/sections/segments of the code and hence commented /* char ans2; cout<<"do u want to check reduce matrix? Press y to check reduce matrix and press any char to skip this"<<endl; cin>>ans2; if(ans2=='y') { int i,j; cout<<"Enter row i and col j to get reduced matrix"<<endl; cin>>i>>j; // i=i-1; // j=j-1; c_obj=reduced(obj,i,j); char ans1; cout<<"Do you want to display the reduced matrix? If yes, Press y "<<endl; cin>>ans1; if(ans1=='y') { cout<<"Displaying reduced matrix..."<<endl; display(c_obj); } }*/ //Find Determinant cout<<"Finding determinant......"<<endl; cout<<"The determinant is"<<determinant(obj)<<endl; //Find Cofactor if user wishes to char ans3; cout<<"Do you want to find cofactor? Press y if yes"<<endl; cin>>ans3; while(ans3=='y') { int i,j; cout<<"Finding cofactor. Enter row and column"<<endl; cin>>i>>j; cout<<"Cofactor of a["<<i<<"]["<<j<<"] is "<<cofactor(obj,i,j)<<endl; cout<<"want of find cofactor of another element? Press y for yes"<<endl; cin>>ans3; } // Following lines were to meant to test ONLY the transpose function and hence commented /* cout<<"Printing Transpose of the matrix "<<endl; matrix trans1; trans1=transpose(obj); display(trans1); */ // Find Inverse cout<<"\n\n\n Finding Inverse. . .\n\n"; matrix obj_inv2; obj_inv2=inverse(obj); display(obj_inv2); // Display the matrix inverse getch(); return 0; } void input(matrix &obj) { // This function gets elements of a matrix input by the user // Parameter is the structure object obj (used throughout the program) // Parameter is "passed by reference" so as to reflect the changes made by this function, to other functions that call it cout<<"Enter the matrix "<<endl; for (int i=1;i<=obj.n;i++) { for (int j=1;j<=obj.n;j++) { cout<<"Enter the element a["<<i<<"]["<<j<<"] : "; cout<<endl; cin>>obj.a[i][j]; } } } void display(matrix &obj) { // This function displays elements of a matrix passed to it as a parameter // Parameter is the structure object obj (used throughout the program) // Parameter is "passed by reference" but may be "passed by value" also. if(obj.n==0) return; else{ cout<<"The matrix is: "<<endl; for (int i=1;i<=obj.n;i++) { for (int j=1;j<=obj.n;j++) { cout<<obj.a[i][j]<<" "; } cout<<endl; } }} matrix reduced(matrix &obj, int i,int j) { // This function reduces the matrix passed as input to it // The 'reduction' requirement is like this: // Eliminate the row i and column j from the given matrix to get the reduced matrix // This is done by the following logic: // a is given matrix. c_obj is desired reduced matrix // i. Using two for loops (iterating with p and q here) as usual, we scan the given matrix. // row and col represent the current location pointer of row and column of the required reduced matrix. // ii. All elements from given matrix are copied to reduced matrix except for those corresponding to // row i and column j // iii. The reduced matrix has its dimensions one less than that of given matrix int row=1,col=1; for(int p=1;p<=obj.n;p++) // outer loop traverses through rows as usual { for(int q=1;q<=obj.n;q++) // inner loop traverses through columns as usual { if((p!=i)&&(q!=j)) // Skip the elements corresponding to row i OR column j of the given matrix { c_obj.a[row][col]=obj.a[p][q]; col=col+1; } if(col>=obj.n) // When column 'col' of reduced matrix reaches (or exceeds n), reset it to 1 { // and increment 'row'. This means current row of reduced matrix got filled and // we need to begin filling a new row. col=1; row=row+1; if (row>=obj.n) //This represents the case when both 'col' and 'row of reduce matrix reach (or // exceed) n. This means the reduced matrix has been filled up.Break out of the loops. break; } } } c_obj.n=obj.n-1; // Fix the dimension of the reduced matrix one less than the given input matrix return c_obj; // Return the reduced matrix to the calling function. } float determinant(matrix obj) { // This function is called recursively until we get dimension = 1 where the only element in the matrix gets returned. float det=0; if(obj.n==1) {return obj.a[1][1]; } else { for(int scan=1;scan<=obj.n;scan++)//Fix the first row and vary the column in this row using for loop iteration variable 'scan' { det=det+obj.a[1][scan]*int(pow(-1,(1+scan)))*determinant(reduced(obj,1,scan)); // det is calculated to be the sum of the following // i. prev value stored in det. // ii. current element in the first row (i.e. a[1][scan]) MULTIPLIED by -1^(i+j) [i is 1 for 1st row and j is nothing but scan here MULTIPLIED by the reduced matrix corresponding to this i (1) and j (scan) // PLEASE UNDERSTAND BY COMPARING THIS WITH THE MATHEMATICAL WAY OF CALCULATING DETERMINANT // - It's computed in a similar way here. } return det; } } float cofactor(matrix obj,int i,int j) { // The computation done here is like this: // If the matrix (passed as paramenter) has dimension = 1, return the only element as cofactor // Else, return determinant of the reduced matrix corresponding to i and j passed with a // multiplication factor -1^(i+j) float cofact; if(obj.n==1) { return obj.a[1][1]; } else { cofact=int(pow(-1,(i+j)))*determinant(reduced(obj,i,j)); } return cofact; } matrix transpose(matrix obj) { // Transpose matrix is the given matrix with its rows and columns interchanged. // Just invert the elements during storing when scanning through the for loops // trans_obj is the transposed matrix, returned by the function. // obj is the input matrix passed to this function. trans_obj.n=obj.n; for(int i=1;i<=obj.n;i++) { for(int j=1;j<=obj.n;j++) { trans_obj.a[i][j]=obj.a[j][i]; } } return trans_obj; } matrix adjoint(matrix obj) { // obj_adj is adjoint matrix and obj_cof is cofactor matrix // both have dimensions n // Cofactor matrix of a given matrix is a matrix whose elements are the cofactors of the respective // elements of the given matrix // Adjoint matrix is transpose of cofactor matrix. Return this obj_adj.n=obj.n; obj_cof.n=obj.n; for(int i=1;i<=obj.n;i++) { for(int j=1;j<=obj.n;j++) { obj_cof.a[i][j]=cofactor(obj,i,j); } } obj_adj=transpose(obj_cof); return obj_adj; } matrix inverse(matrix obj) { // Formula : Inverse of a matrix is = adj(matrix)/its determinant float d=determinant(obj); // First find determinant of the given matrix matrix obj_null; obj_null.n=0; // Display error message if determinant is 0 if(d==0) { cout<<"Inverse can be found only if the determninant of the matrix is non-zero"<<endl; return obj_null; } // Determinant is non-zero - Proceed finding inverse using the above formula else { matrix obj_adj1=adjoint(obj); obj_adj1.n=obj.n; obj_inv.n=obj.n; for(int i=1;i<=obj.n;i++) { for(int j=1;j<=obj.n;j++) { obj_inv.a[i][j]=(obj_adj1.a[i][j])/d; } } return obj_inv; } }

A singular matrix is a matrix that is not invertible. If a matrix is not invertible, then:â€¢ The determinant of the matrix is 0.â€¢ Any matrix multiplied by that matrix doesn't give the identity matrix.There are a lot of examples in which a singular matrix is an idempotent matrix. For instance:M =[1 1][0 0]Take the product of two M's to get the same M, the given!M x M = MSo yes, SOME singular matrices are idempotent matrices! How? Let's take a 2 by 2 identity matrix for instance.I =[1 0][0 1]I x I = I obviously.Then, that nonsingular matrix is also idempotent!Hope this helps!

Related questions

An adjoint is a matrix in which each element is the cofactor of an associated element of another matrix.

You basically write a nested for loop (one for within another one), to copy the elements of the matrix to a new matrix.

C Examples on Matrix OperationsA matrix is a rectangular array of numbers or symbols arranged in rows and columns. The following section contains a list of C programs which perform the operations of Addition, Subtraction and Multiplication on the 2 matrices. The section also deals with evaluating the transpose of a given matrix. The transpose of a matrix is the interchange of rows and columns.The section also has programs on finding the trace of 2 matrices, calculating the sum and difference of two matrices. It also has a C program which is used to perform multiplication of a matrix using recursion.C Program to Calculate the Addition or Subtraction & Trace of 2 MatricesC Program to Find the Transpose of a given MatrixC Program to Compute the Product of Two MatricesC Program to Calculate the Sum & Difference of the MatricesC Program to Perform Matrix Multiplication using Recursion

Complexity of an algorithm is a measure of how long an algorithm would take to complete given

a write the algorithm to concatenate two given string

The matrices that follow d rule of reflexivity is known as ref matrix

This is the definition of an algorithm - a list of orders of how to solve a given programming problem.

MCL refers to the generic MCL algorithm and the MCL process on which the algorithm is based. mcl refers to the implementation, In some places MCL is written where MCL or mcl can be read.mcl is what you use for clustering. It implements the MCL algorithm, which is a cluster algorithm for graphs. The MCL algorithm is basically a shell in which the MCL process is computed and interpreted, The MCL process generates a sequence of stochastic matrices given some initial stochastic matrix. The elements with even index are obtained by expanding the previous element, and the elements with odd index are obtained by inflatingthe previous element given some inflation constant. Expansion is nothing but normal matrix squaring, and inflation is a particular way of rescaling the entries of a stochastic matrix such that it remains stochastic.What do the letters MCL stand for? For Markov Cluster. The MCL algorithm is a cluster algorithm that is basically a shell in which an algebraic process is computed. This process iteratively generates stochastic matrices, also known as Markov matrices, named after the famous Russian mathematician Andrei Markov.What is the MCL and What Does it do? The MCL, when referred to in human anatomy is a ligament in the knee. MCL is the abbreviation for medial collateral ligament. The MCL resists widening of the inside of the knee, (from the weight of our bodies) and prevents it from "opening up".

The generator matrix is made out of that code word and all the possibilities for the code words. The number of rows of the generator matrix are the number of message bits and the number of columns are equal to the total number of bits i.e parity bits + message bits. The only necessary condition is that each row of generator matrix is linearly independent of the other row.

In computer science, deterministic algorithm is an algorithm which, given a particular input, always produces the same result. This is used to increase the efficiency of machines.

It is a basic algorithm for generating lines on computer screen. line is generated between given 2 endpoints

means whether the matrix is same or not program for symmetric matrix : include<stdio.h> #include<conio.h> main() { int a[10][10],at[10][10],k,i,j,m,n; clrscr(); printf("enter the order of matrix"); scanf("%d %d",&m,&n); printf("enter the matrix"); for(i=0;i<m;i++) { for(j=0;j<n;j++) scanf("%d",&a[i][j]); } for(i=0;i<m;i++) { for(j=0;j<n;j++) at[i][j]=a[j][i]; } for(i=0;i<m;i++) { for(j=0;j<n;j++) { if(at[i][j]!=a[i][j]) k=1; } } if(k==1) printf("not symmetric"); else printf("symmetric"); getch(); }