借助矩阵的合同变换法，给出了化实二次型为标准形的方法、求标准正交基的方法，并给出了正定二次型判定定理的新证明。

By means of congruent transformation in matrix, the method of transforming real quadratic form into standard form and the method of normal orthogonal basis are given in this paper.

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在此基础上，利用非负方阵的置换标准形证明了谱分解定理。

Based on this result, the spectral decomposition theorems are proved by using permutation canonical forms of nonnegative matrices.

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