C++ M*N阶矩阵方差、特征值、特征向量、均值求解

要求解一个M*N阶矩阵的方差、特征值、特征向量、均值，可以使用以下步骤：

1. 计算矩阵的均值向量，即将每一列的元素相加并除以元素个数，得到一个N维向量。
2. 将矩阵的每一列减去均值向量，得到一个新的矩阵。
3. 计算新矩阵的协方差矩阵，即将新矩阵乘以其转置矩阵，再除以元素个数。
4. 计算协方差矩阵的特征值和特征向量，可以使用特征值分解或者奇异值分解等方法。
5. 特征值即为方差，特征向量即为矩阵的主成分。可以选择前k个主成分来进行降维处理。

下面是一个C++代码示例：

#include <iostream>
#include <iomanip>
#include <cmath>

using namespace std;

const double EPS = 1e-8;

int n, m;
double a[1010][1010], mean[1010], cov[1010][1010], eig[1010], eigv[1010][1010];

void printMatrix(double mat[][1010], int n, int m) {
    cout << fixed << setprecision(2);
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            cout << mat[i][j] << " ";
        }
        cout << endl;
    }
}

void printVector(double vec[], int n) {
    cout << fixed << setprecision(2);
    for (int i = 0; i < n; i++) {
        cout << vec[i] << " ";
    }
    cout << endl;
}

void calcMean() {
    for (int j = 0; j < m; j++) {
        double sum = 0.0;
        for (int i = 0; i < n; i++) {
            sum += a[i][j];
        }
        mean[j] = sum / n;
    }
}

void calcCov() {
    for (int j = 0; j < m; j++) {
        for (int k = 0; k < m; k++) {
            double sum = 0.0;
            for (int i = 0; i < n; i++) {
                sum += (a[i][j] - mean[j]) * (a[i][k] - mean[k]);
            }
            cov[j][k] = sum / n;
        }
    }
}

bool jacobi(double mat[][1010], int n, double eig[], double eigv[][1010]) {
    const int MAX_ITER = 100;
    const double TOL = 1e-8;

    double b[1010];
    double z[1010];
    double d[1010];

    for (int ip = 0; ip < n; ip++) {
        for (int iq = 0; iq < n; iq++) {
            eigv[ip][iq] = 0.0;
        }
        eigv[ip][ip] = 1.0;
    }

    for (int ip = 0; ip < n; ip++) {
        b[ip] = d[ip] = mat[ip][ip];
        eig[ip] = mat[ip][ip];
        z[ip] = 0.0;
    }

    for (int i = 0; i < MAX_ITER; i++) {
        double sm = 0.0;
        for (int ip = 0; ip < n - 1; ip++) {
            for (int iq = ip + 1; iq < n; iq++) {
                sm += fabs(mat[ip][iq]);
            }
        }

        if (sm < TOL) {
            return true;
        }

        double tresh = (i < 3) ? (0.2 * sm / (n * n)) : 0.0;

        for (int ip = 0; ip < n - 1; ip++) {
            for (int iq = ip + 1; iq < n; iq++) {
                double g = 100.0 * fabs(mat[ip][iq]);

                if (i > 3 && fabs(d[ip]) + g == fabs(d[ip]) && fabs(d[iq]) + g == fabs(d[iq])) {
                    mat[ip][iq] = 0.0;
                } else if (fabs(mat[ip][iq]) > tresh) {
                    double h = d[iq] - d[ip];
                    double t;

                    if (fabs(mat[ip][iq]) < EPS) {
                        t = mat[ip][iq] / h;
                    } else {
                        double theta = 0.5 * h / mat[ip][iq];
                        t = 1.0 / (fabs(theta) + sqrt(1.0 + theta * theta));
                        if (theta < 0.0) {
                            t = -t;
                        }
                    }

                    double c = 1.0 / sqrt(1 + t * t);
                    double s = t * c;
                    double tau = s / (1.0 + c);
                    h = t * mat[ip][iq];
                    z[ip] -= h;
                    z[iq] += h;
                    d[ip] -= h;
                    d[iq] += h;
                    mat[ip][iq] = 0.0;

                    for (int j = 0; j < ip; j++) {
                        g = mat[j][ip];
                        h = mat[j][iq];
                        mat[j][ip] = g - s * (h + g * tau);
                        mat[j][iq] = h + s * (g - h * tau);
                    }

                    for (int j = ip + 1; j < iq; j++) {
                        g = mat[ip][j];
                        h = mat[j][iq];
                        mat[ip][j] = g - s * (h + g * tau);
                        mat[j][iq] = h + s * (g - h * tau);
                    }

                    for (int j = iq + 1; j < n; j++) {
                        g = mat[ip][j];
                        h = mat[iq][j];
                        mat[ip][j] = g - s * (h + g * tau);
                        mat[iq][j] = h + s * (g - h * tau);
                    }

                    for (int j = 0; j < n; j++) {
                        g = eigv[j][ip];
                        h = eigv[j][iq];
                        eigv[j][ip] = g - s * (h + g * tau);
                        eigv[j][iq] = h + s * (g - h * tau);
                    }
                }
            }
        }

        for (int ip = 0; ip < n; ip++) {
            b[ip] += z[ip];
            d[ip] = b[ip];
            z[ip] = 0.0;
        }
    }

    return false;
}

int main() {
    cin >> n >> m;
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < m; j++) {
            cin >> a[i][j];
        }
    }

    calcMean();
    cout << "Mean Vector:" << endl;
    printVector(mean, m);

    calcCov();
    cout << "Covariance Matrix:" << endl;
    printMatrix(cov, m, m);

    jacobi(cov, m, eig, eigv);
    cout << "Eigenvalues:" << endl;
    printVector(eig, m);
    cout << "Eigenvectors:" << endl;
    printMatrix(eigv, m, m);

    return 0;
}

这段代码可以计算一个M*N阶矩阵的均值向量、协方差矩阵、特征值、特征向量。其中，使用了Jacobi方法来进行特征值和特征向量的计算。