## Hokkaido Mathematical Journal

### Maximization of quadratic forms expressed by distance matrices

#### Abstract

If $x,y,z$ are real numbers satisfying $x+y+z=1$, then the maximum of the quadratic form $axy+bxz+cyz$ with positive constants $a,b,c$ is $$\displaystyle{\frac{abc}{2ab+2ac+2bc-a^2-b^2-c^2}}$$ under the assumption $\sqrt{a}<\sqrt{b}+\sqrt{c}$. Extending this fact, we give the maximum of the quadratic form $\displaystyle\sum_{1 \le i< j \le n} a_{ij}x_i x_j$ in $n$-variables $x_1,\ldots,x_n$ satisfying $\displaystyle\sum_{i=1}^{n} x_i = 1$ with constants $a_{ij} \ge 0$ under certain assumptions.

#### Article information

Source
Hokkaido Math. J., Volume 35, Number 3 (2006), 641-658.

Dates
First available in Project Euclid: 29 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1285766422

Digital Object Identifier
doi:10.14492/hokmj/1285766422

Mathematical Reviews number (MathSciNet)
MR2275987

Zentralblatt MATH identifier
1124.15022

Subjects
Primary: 15A63: Quadratic and bilinear forms, inner products [See mainly 11Exx]
Secondary: 15A48

#### Citation

IZUMINO, Saichi; NAKAMURA, Noboru. Maximization of quadratic forms expressed by distance matrices. Hokkaido Math. J. 35 (2006), no. 3, 641--658. doi:10.14492/hokmj/1285766422. https://projecteuclid.org/euclid.hokmj/1285766422