Question

Find the orthogonal canonical reduction of the quadratic form
x2+y2+z2

Answer 1

Answer on Question #79907 – Math – Linear Algebra

Question

Find the orthogonal canonical reduction of the quadratic form $x^{2} + y^{2} + z^{2}$.

Solution

A quadratic form in 3 variables $Q(x,y,z)$ is a function of the form $Q(x,y,z) = q_{xx}x^{2} + 2\cdot q_{xy}xy + 2\cdot q_{xz}xz + q_{yy}y^{2} + 2\cdot q_{yz}xy + q_{zz}z^{2}$. It is known that by changing the system of coordinates $(x,y,z)\rightarrow (x^{\prime},y^{\prime},z^{\prime})$ in some way, any quadratic form can be transformed to a canonical form $Q(x,y,z) = D(x^{\prime},y^{\prime},z^{\prime}) = d_{x^{\prime}x^{\prime}}x^{\prime 2} + d_{z^{\prime}z^{\prime}}y^{\prime 2} + d_{z^{\prime}z^{\prime}}z^{\prime 2}$. The reduction is called orthogonal if the transformation is defined by an orthogonal matrix.

In the case of this problem, the quadratic form is already a linear combination of the squares of the variables. And the trivial transformation is defined by an orthogonal matrix, i.e. the identity matrix.

Answer:

The quadratic form $x^{2} + y^{2} + z^{2}$ is a orthogonal canonical form.

Answer provided by https://www.AssignmentExpert.com