def lambda_sum_largest(X, k): """Sum of the largest k eigenvalues. """ X = Expression.cast_to_const(X) if X.size[0] != X.size[1]: raise ValueError("First argument must be a square matrix.") elif int(k) != k or k <= 0: raise ValueError("Second argument must be a positive integer.") """ S_k(X) denotes lambda_sum_largest(X, k) t >= k S_k(X - Z) + trace(Z), Z is PSD implies t >= ks + trace(Z) Z is PSD sI >= X - Z (PSD sense) which implies t >= ks + trace(Z) >= S_k(sI + Z) >= S_k(X) We use the fact that S_k(X) = sup_{sets of k orthonormal vectors u_i}\sum_{i}u_i^T X u_i and if Z >= X in PSD sense then \sum_{i}u_i^T Z u_i >= \sum_{i}u_i^T X u_i We have equality when s = lambda_k and Z diagonal with Z_{ii} = (lambda_i - lambda_k)_+ """ Z = Semidef(X.size[0]) return k*lambda_max(X - Z) + trace(Z)
def lambda_min(X): """ Minimum eigenvalue; :math:`\\lambda_{\\min}(A)`. """ X = Expression.cast_to_const(X) return -lambda_max(-X)