ChapterZero

Positive definiteness of a certain matrix

Filed under: Mathematics — Alex @ 12:00 pm 6/15/2006

Let $0 < t_1 < \cdots < t_n$ . Then why is 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...

Possibly relevant posts:

Cholesky decomposition (7/11/2006)
Chapter 9 of Principles of Mathematical Analysis (3/22/2005)
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2 Comments »

It’s easier to think of

A-(matrix of all t_1’s),

and use induction.

Comment by George — 6/22/2006 @ 9:13 am
Ahh.. thanks. Neat trick.

Comment by Alex — 6/23/2006 @ 9:15 am

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Positive definiteness of a certain matrix

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