Write the following positive definite quadratic forms as a...

Write the following positive definite quadratic forms as a sum of pure squares, as in (3.79): (a) $16 x_1^2+25 x_2^2$, (b) $x_1^2-2 x_1 x_2+4 x_2^2$, (c) $5 x_1^2+4 x_1 x_2+3 x_2^2$, (d) $3 x_1^2-2 x_1 x_2-2 x_1 x_3+2 x_2^2+6 x_3^2$, (e) $x_1^2+x_1 x_2+x_2^2+x_2 x_3+x_3^2$, (f) $4 x_1^2-2 x_1 x_2-4 x_1 x_3+\frac{1}{2} x_2^2-x_2 x_3+6 x_3^2$, (g) $3 x_1^2+2 x_1 x_2+3 x_2^2+2 x_2 x_3+3 x_3^2+2 x_3 x_4+3 x_4^2$.

Write the following positive definite quadratic forms as a sum of pure squares, as in (3.79):
(a) $16 x_1^2+25 x_2^2$,
(b) $x_1^2-2 x_1 x_2+4 x_2^2$,
(c) $5 x_1^2+4 x_1 x_2+3 x_2^2$,
(d) $3 x_1^2-2 x_1 x_2-2 x_1 x_3+2 x_2^2+6 x_3^2$,
(e) $x_1^2+x_1 x_2+x_2^2+x_2 x_3+x_3^2$,
(f) $4 x_1^2-2 x_1 x_2-4 x_1 x_3+\frac{1}{2} x_2^2-x_2 x_3+6 x_3^2$,
(g) $3 x_1^2+2 x_1 x_2+3 x_2^2+2 x_2 x_3+3 x_3^2+2 x_3 x_4+3 x_4^2$.

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Key Concepts

Completing the Square

This is an algebraic technique used to rewrite a quadratic expression as the square of a binomial plus a constant. In the context of quadratic forms, completing the square helps isolate square terms and can lead to a clearer understanding of the form's structure, facilitating its conversion into a sum of squares.

Positive Definite Quadratic Forms

A quadratic form is positive definite if it takes strictly positive values for all nonzero vectors in its domain. This property guarantees that the quadratic form has a unique minimum and that it can be expressed as a sum of squares of linear forms, which is important for stability analysis and optimization.

Quadratic Forms

A quadratic form is a homogeneous polynomial of degree two in several variables, often written in matrix notation using a symmetric matrix. This representation is fundamental in fields such as linear algebra, optimization, and differential geometry.

Sum of Squares Representation

Expressing a quadratic form as a sum of pure squares means rewriting it in a form where it is a sum of squared linear expressions. This representation simplifies the analysis of the quadratic form by directly relating it to its geometric properties, such as lengths and angles, and is often used in proving properties like positive definiteness.

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