# 查找矩陣的法線和跡線的程序 > 原文： [https://www.geeksforgeeks.org/program-find-normal-trace-matrix/](https://www.geeksforgeeks.org/program-find-normal-trace-matrix/) 給定 2D 矩陣，任務是找到矩陣的“跡線”和“法線”。 **矩陣的標準**定義為矩陣元素平方和的平方根。 n×n 方陣的 [**跡線**](https://en.wikipedia.org/wiki/Trace_(linear_algebra)) 是對角元素的總和。 **示例**： ``` Input : mat[][] = {{7, 8, 9}, {6, 1, 2}, {5, 4, 3}}; Output : Normal = 16 Trace = 11 Explanation : Normal = sqrt(7*7+ 8*8 + 9*9 + 6*6 + 1*1 + 2*2 + 5*5 + 4*4 + 3*3) = 16 Trace = 7+1+3 = 11 Input :mat[][] = {{1, 2, 3}, {6, 4, 5}, {2, 1, 3}}; Output : Normal = 10 Trace = 8 Explanation : Normal = sqrt(1*1 +2*2 + 3*3 + 6*6 + 4*4 + 5*5 + 2*2 + 1*1 + 3*3) Trace = 8(1+4+3) ```  ## C++ ```cpp // C++ program to find trace and normal // of given matrix #include<bits/stdc++.h> using namespace std; // Size of given matrix const int MAX = 100; // Returns Normal of a matrix of size n x n int findNormal(int mat[][MAX], int n) { ????int sum = 0; ????for (int i=0; i<n; i++) ????????for (int j=0; j<n; j++) ????????????sum += mat[i][j]*mat[i][j]; ????return sqrt(sum); } // Returns trace of a matrix of size n x n int findTrace(int mat[][MAX], int n) { ????int sum = 0; ????for (int i=0; i<n; i++) ?????????sum += mat[i][i]; ????return sum; } // Driven source int main() { ????int mat[][MAX] = {{1, 1, 1, 1, 1}, ????????{2, 2, 2, 2, 2}, ????????{3, 3, 3, 3, 3}, ????????{4, 4, 4, 4, 4}, ????????{5, 5, 5, 5, 5}, ????}; ????cout << "Trace of Matrix = " ?????????<< findTrace(mat, 5) << endl; ????cout << "Normal of Matrix = " ?????????<< findNormal(mat, 5) << endl; ????return 0; } ``` ## Java ```java // Java program to find trace and normal // of given matrix import java.io.*; class GFG { // Size of given matrix static? int MAX = 100; // Returns Normal of a matrix of size n x n ?static int findNormal(int mat[][], int n) { ????int sum = 0; ????for (int i=0; i<n; i++) ????????for (int j=0; j<n; j++) ????????????sum += mat[i][j]*mat[i][j]; ????return (int)Math.sqrt(sum); } // Returns trace of a matrix of size n x n ?static int findTrace(int mat[][], int n) { ????int sum = 0; ????for (int i=0; i<n; i++) ????????sum += mat[i][i]; ????return sum; } // Driven source public static void main (String[] args) { ????????????int mat[][] = {{1, 1, 1, 1, 1}, ????????{2, 2, 2, 2, 2}, ????????{3, 3, 3, 3, 3}, ????????{4, 4, 4, 4, 4}, ????????{5, 5, 5, 5, 5}, ????}; ????System.out.println ("Trace of Matrix = " ?????????+ findTrace(mat, 5)); ????System.out.println ("Normal of Matrix = " ?????????+ findNormal(mat, 5)); ????} }? // This code is contributed by vt_m. ``` ## Python3 ```py # Python3 program to find trace and? # normal of given matrix? import math # Size of given matrix? MAX = 100;? # Returns Normal of a matrix? # of size n x n? def findNormal(mat, n):? ????sum = 0;? ????for i in range(n):? ????????for j in range(n):? ????????????sum += mat[i][j] * mat[i][j];? ????return math.floor(math.sqrt(sum));? # Returns trace of a matrix of? # size n x n? def findTrace(mat, n):? ????sum = 0;? ????for i in range(n):? ????????sum += mat[i][i];? ????return sum;? # Driver Code? mat = [[1, 1, 1, 1, 1],? ???????[2, 2, 2, 2, 2],? ???????[3, 3, 3, 3, 3],? ???????[4, 4, 4, 4, 4],? ???????[5, 5, 5, 5, 5]];? print("Trace of Matrix =", findTrace(mat, 5));? print("Normal of Matrix =", findNormal(mat, 5));? # This code is contributed by mits ``` ## C# ```cs // C# program to find trace and normal // of given matrix using System; class GFG { ????// Returns Normal of a matrix of ????// size n x n ????static int findNormal(int [,]mat, int n) ????{ ????????int sum = 0; ????????for (int i = 0; i < n; i++) ????????????for (int j = 0; j < n; j++) ????????????????sum += mat[i,j] * mat[i,j]; ????????return (int)Math.Sqrt(sum); ????} ????// Returns trace of a matrix of size? ????// n x n ????static int findTrace(int [,]mat, int n) ????{ ????????int sum = 0; ????????for (int i = 0; i < n; i++) ????????????sum += mat[i,i]; ????????return sum; ????} ????// Driven source ????public static void Main ()? ????{ ????????int [,]mat = { {1, 1, 1, 1, 1}, ???????????????????????{2, 2, 2, 2, 2}, ???????????????????????{3, 3, 3, 3, 3}, ???????????????????????{4, 4, 4, 4, 4}, ???????????????????????{5, 5, 5, 5, 5}, ????}; ????Console.Write ("Trace of Matrix = " ????????????+ findTrace(mat, 5) + "\n"); ????Console.Write("Normal of Matrix = " ????????????????????+ findNormal(mat, 5)); ????} }? // This code is contributed by nitin mittal. ``` ## PHP ```php <?php // PHP program to find trace and? // normal of given matrix // Size of given matrix $MAX = 100; // Returns Normal of a? // matrix of size n x n function findNormal($mat, $n) { ????$sum = 0; ????for ( $i = 0; $i < $n; $i++) ????????for ( $j = 0; $j < $n; $j++) ????????????$sum += $mat[$i][$j] *? ????????????????????$mat[$i][$j]; ????return floor(sqrt($sum)); } // Returns trace of a // matrix of size n x n function findTrace( $mat, $n) { ????$sum = 0; ????for ( $i = 0; $i < $n; $i++) ????????$sum += $mat[$i][$i]; ????return $sum; } // Driver Code $mat = array(array(1, 1, 1, 1, 1), ?????????????array(2, 2, 2, 2, 2), ?????????????array(3, 3, 3, 3, 3), ?????????????array(4, 4, 4, 4, 4), ?????????????array(5, 5, 5, 5, 5)); echo "Trace of Matrix = ",? ??findTrace($mat, 5),"\n"; echo "Normal of Matrix = " ,? ???????findNormal($mat, 5) ; // This code is contributed by anuj_67\. ?> ``` **輸出**： ``` Trace of matrix = 15 Normal of matrix = 16 ``` **時間復雜度**： O（n * n） **空間復雜度**：`O(1)` **參考**： [https://en.wikipedia.org/wiki/Trace_(linear_algebra）](https://en.wikipedia.org/wiki/Trace_(linear_algebra))