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Check whether a given matrix is orthogonal or not

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  • Difficulty Level : Medium
  • Last Updated : 03 Oct, 2022
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We are given a matrix, we need to check whether it is an orthogonal matrix or not. An orthogonal matrix is a square matrix and satisfies the following condition: 

  A*At = I

Examples : 

Input:  1 0 0
        0 1 0
        0 0 1
Output: Yes
Given Matrix is an orthogonal matrix. When
we multiply it with its transpose, we get
identity matrix.

Input:  1 2 3
        4 5 6
        7 8 9
Output: No
Given Matrix Is Not An Orthogonal Matrix

Simple Solution: The idea is simple, we first find the transpose of matrix. Then we multiply the transpose with the given matrix. Finally, we check if the matrix obtained is identity or not. 

Implementation:

C++




// C++ code to check whether
// a matrix is orthogonal or not
#include <bits/stdc++.h>
using namespace std;
 
#define MAX 100
 
bool isOrthogonal(int a[][MAX],
                  int m, int n)
{
if (m != n)
    return false;
 
// Find product of a[][]
// and its transpose
int prod[n][n];
for (int i = 0; i < n; i++)
{
    for (int j = 0; j < n; j++)
    {
 
    int sum = 0;
    for (int k = 0; k < n; k++)
    {
 
        // Since we are multiplying with
        // transpose of itself. We use
        sum = sum + (a[i][k] * a[j][k]);
    }
 
    prod[i][j] = sum;
    }
}
 
// Check if product is identity matrix
for (int i = 0; i < n; i++)
{
    for (int j = 0; j < n; j++)
    {
    if (i != j && prod[i][j] != 0)
        return false;
    if (i == j && prod[i][j] != 1)
        return false;
    }
}
 
return true;
}
 
// Driver Code
int main()
{
 
int a[][MAX] = {{1, 0, 0},
                {0, 1, 0},
                {0, 0, 1}};
if (isOrthogonal(a, 3, 3))
    cout << "Yes";
else
    cout << "No";
return 0;
}

C




// C code to check whether
// a matrix is orthogonal or not
#include <stdio.h>
#include <stdbool.h>
 
#define MAX 100
 
bool isOrthogonal(int a[][MAX], int m, int n)
{
if (m != n)
    return false;
 
// Find transpose
int trans[n][n];
for (int i = 0; i < n; i++)
    for (int j = 0; j < n; j++)
    trans[i][j] = a[j][i];
 
// Find product of a[][]
// and its transpose
int prod[n][n];
for (int i = 0; i < n; i++)
{
    for (int j = 0; j < n; j++)
    {
 
    int sum = 0;
    for (int k = 0; k < n; k++)
    {
 
        // Since we are multiplying with
        // transpose of itself. We use
        sum = sum + (a[i][k] * a[j][k]);
    }
 
    prod[i][j] = sum;
    }
}
 
// Check if product is identity matrix
for (int i = 0; i < n; i++)
{
    for (int j = 0; j < n; j++)
    {
    if (i != j && prod[i][j] != 0)
        return false;
    if (i == j && prod[i][j] != 1)
        return false;
    }
}
 
return true;
}
 
// Driver Code
int main()
{
 
int a[][MAX] = {{1, 0, 0},
                {0, 1, 0},
                {0, 0, 1}};
if (isOrthogonal(a, 3, 3))
    printf("Yes");
else
    printf("No");;
return 0;
}
 
// This code is contributed by kothavvsaakash.

Java




// Java code to check whether
// a matrix is orthogonal or not
import java .io.*;
class GFG {
     
    //static int MAX =100;
    static boolean isOrthogonal(int [][]a,
                                int m,
                                int n)
    {
        if (m != n)
            return false;
         
        // Find transpose
        int [][]trans = new int[n][n];
        for (int i = 0; i < n; i++)
            for (int j = 0; j < n; j++)
                trans[i][j] = a[j][i];
         
        // Find product of a[][]
        // and its transpose
        int [][]prod = new int[n][n];
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
         
                int sum = 0;
                for (int k = 0; k < n; k++)
                {
                    // Since we are multiplying
                    // transpose of itself. We use
                    sum = sum + (a[i][k] * a[j][k]);
                }
             
                prod[i][j] = sum;
            }
        }
         
        // Check if product is
        // identity matrix
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                if (i != j && prod[i][j] != 0)
                    return false;
                if (i == j && prod[i][j] != 1)
                    return false;
            }
        }
         
        return true;
    }
     
    // Driver code
    static public void main (String[] args)
    {
        int [][]a = {{1, 0, 0},
                    {0, 1, 0},
                    {0, 0, 1}};
        if (isOrthogonal(a, 3, 3))
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}
 
// This code is contributed by anuj_67.

Python3




# Python code to check
# whether a matrix is
# orthogonal or not
 
def isOrthogonal(a, m, n) :
    if (m != n) :
        return False
     
    trans = [[0 for x in range(n)]
                for y in range(n)]
                 
    # Find transpose
    for i in range(0, n) :
        for j in range(0, n) :
            trans[i][j] = a[j][i]
             
    prod = [[0 for x in range(n)]
               for y in range(n)]
                
    # Find product of a[][]
    # and its transpose
    for i in range(0, n) :
        for j in range(0, n) :
     
            sum = 0
            for k in range(0, n) :
         
                # Since we are multiplying
                # with transpose of itself.
                # We use
                sum = sum + (a[i][k] *
                             a[j][k])
     
            prod[i][j] = sum
 
    # Check if product is
    # identity matrix
    for i in range(0, n) :
        for j in range(0, n) :
 
            if (i != j and prod[i][j] != 0) :
                return False
            if (i == j and prod[i][j] != 1) :
                return False
 
    return True
 
# Driver Code
a = [[1, 0, 0],
    [0, 1, 0],
    [0, 0, 1]]
         
if (isOrthogonal(a, 3, 3)) :
    print ("Yes")
else :
    print ("No")
 
# This code is contributed by
# Manish Shaw(manishshaw1)

C#




// C# code to check whether
// a matrix is orthogonal or not
using System;
 
class GFG
{
     
    //static int MAX =100;
    static bool isOrthogonal(int [,]a,
                             int m,
                             int n)
    {
        if (m != n)
            return false;
         
        // Find transpose
        int [,]trans = new int[n, n];
        for (int i = 0; i < n; i++)
            for (int j = 0; j < n; j++)
                trans[i, j] = a[j, i];
         
        // Find product of a[][]
        // and its transpose
        int [,]prod = new int[n, n];
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
         
                int sum = 0;
                for (int k = 0; k < n; k++)
                {
                    // Since we are multiplying
                    // transpose of itself. We use
                    sum = sum + (a[i, k] * a[j, k]);
                }
             
                prod[i, j] = sum;
            }
        }
         
        // Check if product is
        // identity matrix
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                if (i != j && prod[i, j] != 0)
                    return false;
                if (i == j && prod[i, j] != 1)
                    return false;
            }
        }
         
        return true;
    }
     
    // Driver code
    static public void Main ()
    {
        int [,]a = {{1, 0, 0},
                    {0, 1, 0},
                    {0, 0, 1}};
        if (isOrthogonal(a, 3, 3))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}
 
// This code is contributed by vt_m

PHP




<?php
// PHP code to check whether
// a matrix is orthogonal or not
 
function isOrthogonal($a,
                      $m, $n)
{
    if ($m != $n)
        return false;
     
    // Find transpose
    for($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $n; $j++)
        $trans[$i][$j] = $a[$j][$i];
     
    // Find product of a[][]
    // and its transpose
    for($i = 0; $i < $n; $i++)
    {
        for ($j = 0; $j < $n; $j++)
        {
     
        $sum = 0;
        for ($k = 0; $k < $n; $k++)
        {
     
            // Since we are multiplying with
            // transpose of itself. We use
            $sum = $sum + ($a[$i][$k] *
                           $a[$j][$k]);
        }
     
        $prod[$i][$j] = $sum;
        }
    }
     
    // Check if product is
    // identity matrix
    for($i = 0; $i < $n; $i++)
    {
        for ($j = 0; $j < $n; $j++)
        {
            if ($i != $j && $prod[$i][$j] != 0)
                return false;
            if ($i == $j && $prod[$i][$j] != 1)
                return false;
        }
    }
     
    return true;
}
 
    // Driver Code
    $a = array(array(1, 0, 0),
               array(0, 1, 0),
               array(0, 0, 1));
    if (isOrthogonal($a, 3, 3))
        echo "Yes";
    else
        echo "No";
 
// This code is contributed by ajit.
?>

Javascript




<script>
 
// Javascript code to check whether
// a matrix is orthogonal or not
 
// static int MAX =100;
function isOrthogonal(a, m, n)
{
    if (m != n)
        return false;
      
    // Find transpose
    let trans = new Array(n);
     
    for(let i = 0; i < n; i++)
    {
        trans[i] = new Array(n);
        for(let j = 0; j < n; j++)
            trans[i][j] = a[j][i];
    }
      
    // Find product of a[][]
    // and its transpose
    let prod = new Array(n);
    for(let i = 0; i < n; i++)
    {
        prod[i] = new Array(n);
        for(let j = 0; j < n; j++)
        {
            let sum = 0;
            for(let k = 0; k < n; k++)
            {
                 
                // Since we are multiplying
                // transpose of itself. We use
                sum = sum + (a[i][k] * a[j][k]);
            }
            prod[i][j] = sum;
        }
    }
      
    // Check if product is
    // identity matrix
    for(let i = 0; i < n; i++)
    {
        for(let j = 0; j < n; j++)
        {
            if (i != j && prod[i][j] != 0)
                return false;
            if (i == j && prod[i][j] != 1)
                return false;
        }
    }
    return true;
}
 
// Driver code
let a = [ [ 1, 0, 0 ],
          [ 0, 1, 0 ],
          [ 0, 0, 1 ] ];
           
if (isOrthogonal(a, 3, 3))
  document.write("Yes");
else
  document.write("No");
   
// This code is contributed by divyesh072019
 
</script>

Output : 

Yes

 

Time Complexity: O(n*n*n)
Auxiliary Space: O(n*n)

An efficient solution is to combine three traversals into one. Instead of explicitly finding transpose, we use a[j][k] instead of a[k][j]. Also, instead of explicitly computing the product, we check identity while computing the product. 

Implementation:

C++




// C++ code to check whether
// a matrix is orthogonal or not
#include <bits/stdc++.h>
using namespace std;
 
#define MAX 100
 
bool isOrthogonal(int a[][MAX],
                  int m, int n)
{
if (m != n)
    return false;
 
// Multiply A*A^t
for (int i = 0; i < n; i++)
{
    for (int j = 0; j < n; j++)
    {
 
    int sum = 0;
    for (int k = 0; k < n; k++)
    {
 
        // Since we are multiplying with
        // transpose of itself. We use
        // a[j][k] instead of a[k][j]
        sum = sum + (a[i][k] * a[j][k]);
    }
 
    if (i == j && sum != 1)
        return false;
    if (i != j && sum != 0)
        return false;
    }
}
 
return true;
}
 
// Driver Code
int main()
{
 
int a[][MAX] = {{1, 0, 0},
                {0, 1, 0},
                {0, 0, 1}};
if (isOrthogonal(a, 3, 3))
    cout << "Yes";
else
    cout << "No";
return 0;
}

C




// C code to check whether
// a matrix is orthogonal or not
#include <stdio.h>
#include <stdbool.h>
 
#define MAX 100
 
bool isOrthogonal(int a[][MAX], int m, int n)
{
  if (m != n)
    return false;
 
  // Multiply A*A^t
  for (int i = 0; i < n; i++)
  {
    for (int j = 0; j < n; j++)
    {
 
      int sum = 0;
      for (int k = 0; k < n; k++)
      {
 
        // Since we are multiplying with
        // transpose of itself. We use
        // a[j][k] instead of a[k][j]
        sum = sum + (a[i][k] * a[j][k]);
      }
 
      if (i == j && sum != 1)
        return false;
      if (i != j && sum != 0)
        return false;
    }
  }
 
  return true;
}
 
// Driver Code
int main()
{
 
  int a[][MAX] = {{1, 0, 0},
                  {0, 1, 0},
                  {0, 0, 1}};
  if (isOrthogonal(a, 3, 3))
    printf("Yes");
  else
    printf("No");
  return 0;
}
 
// This code is contributed by kothavvsaakash.

Java




// C# code to check whether
// a matrix is orthogonal or not
using System;
 
class GFG
{
    //static int MAX =100;
 
    static bool isOrthogonal(int[,]a,
                             int m,
                             int n)
    {
        if (m != n)
            return false;
         
        // Multiply A*A^t
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
         
            int sum = 0;
            for (int k = 0; k < n; k++)
            {
         
                // Since we are multiplying
                // with transpose of itself.
                // We use a[j][k] instead
                // of a[k][j]
                sum = sum + (a[i, k] * a[j, k]);
            }
         
            if (i == j && sum != 1)
                return false;
            if (i != j && sum != 0)
                return false;
            }
        }
         
        return true;
    }
     
    // Driver code
    public static void Main ()
    {
        int [,]a = {{1, 0, 0},
                    {0, 1, 0},
                    {0, 0, 1}};
        if (isOrthogonal(a, 3, 3))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
         
    }
}
 
// This code is contributed by vt_m

Python3




# Python code to check
# whether a matrix is
# orthogonal or not
 
def isOrthogonal(a, m, n) :
    if (m != n) :
        return False
     
    # Multiply A*A^t
    for i in range(0, n) :
        for j in range(0, n) :
            sum = 0
            for k in range(0, n) :
         
                # Since we are multiplying
                # with transpose of itself.
                # We use a[j][k] instead
                # of a[k][j]
                sum = sum + (a[i][k] *
                             a[j][k])
         
        if (i == j and sum != 1) :
            return False
        if (i != j and sum != 0) :
            return False
 
    return True
 
# Driver Code
a = [[1, 0, 0],
     [0, 1, 0],
     [0, 0, 1]]
if (isOrthogonal(a, 3, 3)) :
    print ("Yes")
else :
    print ("No")
     
# This code is contributed by
# Manish Shaw(manishshaw1)

C#




// C# code to check whether
// a matrix is orthogonal or not
using System;
 
public class GFG
{
    //static int MAX =100;
 
    static bool isOrthogonal(int[,]a, int m, int n)
    {
        if (m != n)
            return false;
         
        // Multiply A*A^t
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
         
            int sum = 0;
            for (int k = 0; k < n; k++) {
         
                // Since we are multiplying with
                // transpose of itself. We use
                // a[j][k] instead of a[k][j]
                sum = sum + (a[i,k] * a[j,k]);
            }
         
            if (i == j && sum != 1)
                return false;
            if (i != j && sum != 0)
                return false;
            }
        }
         
        return true;
    }
     
    // Driver code   
    public static void Main ()
    {
                 
        int [,]a = {{1, 0, 0},
                    {0, 1, 0},
                    {0, 0, 1}};
        if (isOrthogonal(a, 3, 3))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
         
    }
}
 
// This code is contributed by vt_m

PHP




<?php
// PHP code to check whether
// a matrix is orthogonal or not
 
 
//$MAX = 100;
 
function isOrthogonal($a, $m, $n)
{
if ($m != $n)
    return false;
 
// Multiply A*A^t
for ( $i = 0; $i < $n; $i++)
{
    for ( $j = 0; $j < $n; $j++)
    {
 
    $sum = 0;
    for ( $k = 0; $k < $n; $k++)
    {
 
        // Since we are multiplying
        // with transpose of itself.
        // We use a[j][k] instead
        // of a[k][j]
        $sum = $sum + ($a[$i][$k] *
                       $a[$j][$k]);
    }
 
    if ($i == $j and $sum != 1)
        return false;
    if ($i != $j and $sum != 0)
        return false;
    }
}
 
return true;
}
 
// Driver Code
$a = array(array(1, 0, 0),
           array(0, 1, 0),
           array(0, 0, 1));
if (isOrthogonal($a, 3, 3))
    echo "Yes";
else
    echo "No";
 
// This code is contributed by anuj_67.
?>

Javascript




<script>
 
// Javascript code to check whether
// a matrix is orthogonal or not
 
MAX=100
 
function isOrthogonal(a, m, n)
{
if (m != n)
    return false;
 
// Multiply A*A^t
for (var i = 0; i < n; i++)
{
    for (var j = 0; j < n; j++)
    {
 
    var sum = 0;
    for (var k = 0; k < n; k++)
    {
 
        // Since we are multiplying with
        // transpose of itself. We use
        // a[j][k] instead of a[k][j]
        sum = sum + (a[i][k] * a[j][k]);
    }
 
    if (i == j && sum != 1)
        return false;
    if (i != j && sum != 0)
        return false;
    }
}
 
return true;
}
 
// Driver Code
var a = [[1, 0, 0],
                [0, 1, 0],
                [0, 0, 1]];
if (isOrthogonal(a, 3, 3))
    document.write( "Yes");
else
    document.write( "No");
 
// This code is contributed by noob2000.
</script>

Output : 

Yes

 

Time Complexity: O(n*n*n)
Auxiliary Space: O(1)


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