Disclosed is a transition metal composite hydroxide which contains SO4, a transition metal element, and an alkaline metal other than Li, wherein the ratio of the molar content of the aforementioned alkaline metal to the molar content of the aforementioned SO4 is at least 0.05 but less than 2; also disclosed is a lithium composite metal oxide obtained by maintaining at 650-1000 DEG C and sintering a mix of the aforementioned transition metal composite hydroxide and a lithium compound. Non-aqueous electrolyte secondary cells using the disclosed lithium composite metal oxide as a positive electrode active substance have superior capacity and output characteristics compared to conventional lithium secondary cells, and in particular, are extremely useful in applications demanding a high capacity and a high output during a high current rate, that is, for automobiles and power tools.