前面所说的都是只有二位的数据，现在我们把问题普遍化，将高维数据微量低维数据映射的问题。
假设样本X的数量有m个，特征数目有k个，也就是m*k大小的样本，希望降维之后变成m*n,也就是说，降维到n维的空间内。简单地说，我们需要找前n个主成分。

用矩阵表示如下：

下面用代码封装一个降维工具：

class PCA:
    def __init__(self, n_components):
        '''initiate PCA'''
        assert n_cimponents >= 1, "n_components must be valid"
        self.n_components = n_components
        self.components_ = None
    def fit(self, X, eta=0.01, n_iters=1e4):
        '''get first n pca of data X'''
        assert self.n_components <= X.shape[1],\
            '''n_components must not be greater than the feature number of X'''
        def demean(X):
            return X-np.mean(X,axis=0)
        def f(w,X):
            '''cost function'''
            return np.sum(X.dot(w)**2)/len(X)
        def df(w,X):
            '''derived function'''
            return X.T.dot(X.dot(w))*2./len(X)
        def direction(w):
            '''turn w to unit vector'''
            return w/np.linalg.norm(w)
        def first_component(X, initial_w, eta=0.01, n_iters=1e4,epsilon=1e-8):
            w = direction(initial_w)
            cur_iter=0
            while cur_iter<n_iters:
                gradient=df(w,X)
                last_w = w
                w = w+eta*gradient
                w = direction(w)
                if (abs(f(w,X)-f(last_w,X))<epsilon):
                    break
                cur_iter+=1
            return w
        X_pca = demean(X)
        self.components_ = np.empty(shape=(self.n_components,X.shape[1]))#initiate an empty w matix (n,m)
        for i in range(self.n_components):
            initial_w = np.random.random(X_pca.shape[1])
            w = first_component(X_pca,initial_w, eta, n_iters)
            self.components_[i,:] = w #put w in w matrix
            X_pca = X_pca - X_pca.dot(w).reshape(-1,1)*w
        return self
    def transform(self, X):
        assert X.shape[1]==self.components_.shape[1]
        return X.dot(self.components_.T)
    def inverse_transform(self,X):
        assert X.shape[1]==self.components_.shape[0]
        returnX.dot(self.components_)
    def __repr__(self):
        return "PCA(n_components=%d)"%self.n_components