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%% Basis for Full Rank Matrix % Calculate and verify the orthonormal basis vectors for the range of a % full rank matrix. %% % Define a matrix and find the rank. A = [1 0 1;-1 -2 0; 0 1 -1]; r = rank(A) %% % Since |A| is a square matrix of full rank, the orthonormal basis % calculated by |orth(A)| matches the matrix |U| calculated in the singular % value decomposition, |[U,S] = svd(A,'econ')|. This is because the % singular values of |A| are all nonzero. %% % Calculate the orthonormal basis for the range of |A| using |orth|. Q = orth(A) %% % The number of columns in |Q| is equal to |rank(A)|. Since |A| is of full % rank, |Q| and |A| are the same size. %% % Verify that the basis, |Q|, is orthogonal and normalized within a % reasonable error range. E = norm(eye(r)-Q'*Q,'fro') %% % The error is on the order of |eps|.