# Cholesky Decomposition : Matrix Decomposition

In linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions. One of them is Cholesky Decomposition.

The Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. The Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form A = [L][L]T, where L is a lower triangular matrix with real and positive diagonal entries, and LT denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.

Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose: A = L LT

The following formulas are obtained by solving above lower triangular matrix and its transpose. These are the basis of Cholesky Decomposition Algorithm :

Example :

Input :

Output :



## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Below is the implementation of Cholesky Decomposition.

## C++

 // CPP program to decompose a matrix using  // Cholesky Decomposition  #include  using namespace std;     const int MAX = 100;     void Cholesky_Decomposition(int matrix[][MAX],                                         int n)  {      int lower[n][n];      memset(lower, 0, sizeof(lower));         // Decomposing a matrix into Lower Triangular      for (int i = 0; i < n; i++) {          for (int j = 0; j <= i; j++) {              int sum = 0;                 if (j == i) // summation for diagnols              {                  for (int k = 0; k < j; k++)                      sum += pow(lower[j][k], 2);                  lower[j][j] = sqrt(matrix[j][j] -                                          sum);              } else {                     // Evaluating L(i, j) using L(j, j)                  for (int k = 0; k < j; k++)                      sum += (lower[i][k] * lower[j][k]);                  lower[i][j] = (matrix[i][j] - sum) /                                        lower[j][j];              }          }      }         // Displaying Lower Triangular and its Transpose      cout << setw(6) << " Lower Triangular"            << setw(30) << "Transpose" << endl;      for (int i = 0; i < n; i++) {                     // Lower Triangular          for (int j = 0; j < n; j++)              cout << setw(6) << lower[i][j] << "\t";          cout << "\t";                     // Transpose of Lower Triangular          for (int j = 0; j < n; j++)              cout << setw(6) << lower[j][i] << "\t";          cout << endl;      }  }     // Driver Code  int main()  {      int n = 3;      int matrix[][MAX] = { { 4, 12, -16 },                          { 12, 37, -43 },                          { -16, -43, 98 } };      Cholesky_Decomposition(matrix, n);      return 0;  }

## Java

 // Java program to decompose   // a matrix using Cholesky  // Decomposition     class GFG  {         //static int MAX = 100;  static void Cholesky_Decomposition(int[][] matrix,                                  int n)  {      int[][] lower = new int[n][n];         // Decomposing a matrix       // into Lower Triangular      for (int i = 0; i < n; i++)      {          for (int j = 0; j <= i; j++)          {              int sum = 0;                 // summation for diagnols              if (j == i)               {                  for (int k = 0; k < j; k++)                      sum += (int)Math.pow(lower[j][k], 2);                  lower[j][j] = (int)Math.sqrt(matrix[j][j] -                                                      sum);              }                              else             {                     // Evaluating L(i, j)                   // using L(j, j)                  for (int k = 0; k < j; k++)                      sum += (lower[i][k] *                               lower[j][k]);                  lower[i][j] = (matrix[i][j] - sum) /                                      lower[j][j];              }          }      }         // Displaying Lower       // Triangular and its Transpose      System.out.println(" Lower Triangular\t Transpose");      for (int i = 0; i < n; i++)       {                     // Lower Triangular          for (int j = 0; j < n; j++)              System.out.print(lower[i][j] + "\t");          System.out.print("");                     // Transpose of           // Lower Triangular          for (int j = 0; j < n; j++)              System.out.print(lower[j][i] + "\t");          System.out.println();      }  }     // Driver Code  public static void main(String[] args)  {      int n = 3;      int[][] matrix = new int[][]{{4, 12, -16},                      {12, 37, -43},                      {-16, -43, 98}};                             Cholesky_Decomposition(matrix, n);  }  }     // This code is contributed by mits

## Python3

 # Python3 program to decompose   # a matrix using Cholesky   # Decomposition  import math  MAX = 100;     def Cholesky_Decomposition(matrix, n):         lower = [[0 for x in range(n + 1)]                   for y in range(n + 1)];         # Decomposing a matrix      # into Lower Triangular      for i in range(n):           for j in range(i + 1):               sum1 = 0;                 # sum1mation for diagnols              if (j == i):                   for k in range(j):                      sum1 += pow(lower[j][k], 2);                  lower[j][j] = int(math.sqrt(matrix[j][j] - sum1));              else:                                     # Evaluating L(i, j)                  # using L(j, j)                  for k in range(j):                      sum1 += (lower[i][k] *lower[j][k]);                  if(lower[j][j] > 0):                      lower[i][j] = int((matrix[i][j] - sum1) /                                                 lower[j][j]);         # Displaying Lower Triangular      # and its Transpose      print("Lower Triangular\t\tTranspose");      for i in range(n):                      # Lower Triangular          for j in range(n):              print(lower[i][j], end = "\t");          print("", end = "\t");                     # Transpose of          # Lower Triangular          for j in range(n):              print(lower[j][i], end = "\t");          print("");     # Driver Code  n = 3;  matrix = [[4, 12, -16],            [12, 37, -43],            [-16, -43, 98]];  Cholesky_Decomposition(matrix, n);     # This code is contributed by mits

## C#

 // C# program to decompose   // a matrix using Cholesky  // Decomposition  using System;     class GFG  {         //static int MAX = 100;  static void Cholesky_Decomposition(int[,] matrix,                                     int n)  {      int[,] lower = new int[n, n];         // Decomposing a matrix       // into Lower Triangular      for (int i = 0; i < n; i++)      {          for (int j = 0; j <= i; j++)          {              int sum = 0;                 // summation for diagnols              if (j == i)               {                  for (int k = 0; k < j; k++)                      sum += (int)Math.Pow(lower[j, k], 2);                  lower[j, j] = (int)Math.Sqrt(matrix[j, j] -                                                        sum);              }                              else             {                     // Evaluating L(i, j)                   // using L(j, j)                  for (int k = 0; k < j; k++)                      sum += (lower[i, k] *                               lower[j, k]);                  lower[i, j] = (matrix[i,j] - sum) /                                         lower[j, j];              }          }      }         // Displaying Lower       // Triangular and its Transpose      Console.WriteLine("  Lower Triangular\t   Transpose");      for (int i = 0; i < n; i++)       {                     // Lower Triangular          for (int j = 0; j < n; j++)              Console.Write(lower[i, j] + "\t");          Console.Write("");                     // Transpose of           // Lower Triangular          for (int j = 0; j < n; j++)              Console.Write(lower[j, i] + "\t");          Console.WriteLine();      }  }     // Driver Code  static int Main()  {      int n = 3;      int[,] matrix = {{4, 12, -16},                       {12, 37, -43},                       {-16, -43, 98}};                              Cholesky_Decomposition(matrix, n);      return 0;  }  }     // This code is contributed by mits

## PHP

 

Output:


Lower Triangular                   Transpose
2         0         0             2         6        -8
6         1         0             0         1         5
-8         5         3             0         0         3


References: Wikipedia – Cholesky decomposition
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