Positive definiteness of a certain matrix

June 15th, 2006 ~ Posted in: Mathematics

Let 0 < t_1 < \cdots < t_n . Then why is A positive definite when A_{ij} = \min(t_i, t_j) = t_{\min(i,j)}? I think it is ‘because’
\displaystyle A = \begin{pmatrix} t_1 & t_1 & \ldots & t_1 \\ t_1 & t_2 & \ldots & t_2 \\ & & \vdots & \\ t_1 & t_2 & \ldots & t_n \end{pmatrix}

can be written as the sum of an upper triangular matrix that is positive definite and a strictly lower triangular matrix that is positive semidefinite. But then I have to prove those statements…

Mathematics ~

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