Lecture 149:- Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. In other words, for a symmetric matrix A, A[i][j] = A[j][i] for all i and j. The elements above and below the main diagonal are mirror images of each other.

To represent a symmetric matrix in memory, we can use either a 2D array or a 1D array. Here, we'll discuss the row-major mapping for a symmetric matrix using a 1D array.

In row-major mapping for a symmetric matrix, we store the elements row by row, including the diagonal elements, in a 1D array. The mapping formula to access an element at row i and column j in the 1D array is:

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index = (i * (i + 1) / 2) + j

Here, index is the position of the element in the 1D array, i is the row number (0-based), and j is the column number (0-based).

Let's create a C++ program to demonstrate the row-major mapping for a symmetric matrix:

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#include <iostream> #include <vector> class SymmetricMatrix { private:    int size;                    // Size of the matrix    std::vector<int> elements;   // 1D vector to store matrix elements public:    // Constructor to create a symmetric matrix of given size    SymmetricMatrix(int n) {        size = n;        int totalElements = n * (n + 1) / 2;        elements.resize(totalElements, 0);    }    // Function to set an element at (i, j) position in the symmetric matrix    void setElement(int i, int j, int value) {        if (i < size && j < size) {            int index = (i * (i + 1) / 2) + j;            elements[index] = value;            elements[j * (j + 1) / 2 + i] = value; // Set the corresponding mirrored element        } else {            std::cout << "Invalid index. Element not set." << std::endl;        }    }    // Function to get an element from (i, j) position in the symmetric matrix    int getElement(int i, int j) {        if (i < size && j < size) {            int index = (i * (i + 1) / 2) + j;            return elements[index];        } else {            std::cout << "Invalid index. Returning 0." << std::endl;            return 0;        }    }    // Function to display the symmetric matrix    void displaySymmetricMatrix() {        std::cout << "Symmetric Matrix:" << std::endl;        for (int i = 0; i < size; i++) {            for (int j = 0; j < size; j++) {                std::cout << getElement(i, j) << " ";            }            std::cout << std::endl;        }    } }; int main() {    int size = 4;    SymmetricMatrix matrix(size);    matrix.setElement(0, 0, 1);    matrix.setElement(0, 1, 2);    matrix.setElement(0, 2, 3);    matrix.setElement(0, 3, 4);    matrix.setElement(1, 1, 5);    matrix.setElement(1, 2, 6);    matrix.setElement(1, 3, 7);    matrix.setElement(2, 2, 8);    matrix.setElement(2, 3, 9);    matrix.setElement(3, 3, 10);    matrix.displaySymmetricMatrix();    return 0; }

Output:

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Symmetric Matrix: 1 2 3 4 2 5 6 7 3 6 8 9 4 7 9 10

In this program, we have defined the SymmetricMatrix class, which uses a 1D vector to store the elements of the symmetric matrix. The setElement function sets the element at the (i, j) position by setting both the corresponding element and its mirror element. The getElement function retrieves the element at the (i, j) position. The displaySymmetricMatrix function displays the symmetric matrix with its meaningful elements.

You can set elements and access values using the (i, j) coordinates, and the class will handle the correct mapping to the 1D array using row-major mapping.