In the present paper, we characterize the classes of all triangular matrices, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(|\bar N_p |,|\bar N_q^\theta |_k )$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(|\bar N_p^\theta |_k ,|\bar N_q |)$ \end{document} for the case k ≧ 1, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $|\bar N_p^\theta |_k = \left\{ {a = (a_n ):\sum\limits_{n = 1}^\infty {\theta _n^{k - 1} } \left| {\frac{{p_n }}{{p_n p_{n - 1} }}\sum\limits_{v = 1}^n {P_{v - 1} a_v } } \right|^k < \infty } \right\},$ \end{document} i.e., the set of series summable by absolute weighted mean summability method, and so extend the some well known results.