What are the applications of reducing Quadratic form to canonical form?
One of the applications is determining the type of the second derivative of a C^(2) function f at a critical point a of f. This yields the type of that critical point: it is a (local or global) max point, or a (local or global) min point, or a it is not an extreme point. In some rare cases, we cannot decide whether a is an extreme point or not for f. Reducing to canonical form might be of some help, especially for global max or min points.
Octave is right!
In addition, we can also in elegant way
distinct characteristic roots,
determine definiteness of the quadratic form.
Greetings, Tadeusz
From the canonical form we can easily determine the rank, determinant, trace, characteristic roots, nature of the associated matrix of the quadratic form.
The canonical form will determine the radical, and the Witt index (if the field is finite). You can also easily read off certain maximal singular subspaces, hyperbolic pairs and information of geometric significance.
The local geometry of the universe is driven by the Einstein equations using tensors (gravitation effects on space and time) : the eigenvalues of the local tensor (a differential form in 4 variables to simplify) are these of a quadratic form.
The canonical form helps understanding any change of variables.
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