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In Shilov's Linear Algebra he gives ($\S$7.5) a method for orthogonalization of a symmetric bilinear form (or if you prefer, a quadratic form) which he calls Jacobi's method. In his terminology, the problem is finding a canonical basis for the bilinear form, such that the matrix of the form is diagonal. The only reference I've been able to find outside of Shilov is from the Encyclopedia of Mathematics, which cites another Russian text. This isn't common terminology in the Western world, but a very similar method appears to be the Gram-Schmidt process, though I haven't looked into it as closely, and it applies particularly to Euclidean space and obtains orthonormalization.

The problem is essentially this, in Shilov's terminology. Given a nonsingular symmetric bilinear form $A(x,y)$ acting in a linear space $\mathbf{K}_n$ of dimension $n$ (finite) over a field $K$, and given an arbitrary basis $\{e\} = e_1, e_2, ..., e_n$, such that $A(e_i,e_j) = A(e_j,e_i)$ for all $i,j$, find a transformation to a new basis $\{f\}=f_1,f_2,...,f_n$ such that $A(f_i,f_j)=0$ when $i\neq j$, and $A(e_i,e_i) = \lambda_i$, where $\lambda_i$ is called a canonical coefficient. The basis $\{f\}$ is called a canonical basis of the bilinear form $A(x,y)$.

This problem can be solved by induction, but we wish to find a method such that "the components of the vectors of the canonical basis and the corresponding canonical coefficients $\lambda_i$ [are able] to be determined directly from a knowledge of the elements of the matrix $A_{(f)}$," for $A_{(f)}$ is the matrix of $A(x,y)$ in the basis $\{f\}$. The required method, Shilov goes on to say, is Jacobi's method, as described in the Encyclopedia article above.

How is this method presented in modern literature? What terminology does it goes by?

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Let q be a quadratic form on a space V, represented in a basis $(\vec e_1, ... , \vec e_n)$ by the matrix A, namely $$q(\vec x) = \vec x^t A \vec x$$ Then, Jacobi's method implies that in some basis $(\vec b_1,...,\vec b_n)$, q is diagonal, and for any $\vec x = x'_1\vec b_1+...+x'_n \vec b_n \in V$ $$q(\vec x) = \sum_{i=1}^n\frac{\Delta _{i-1}}{\Delta_i} (x_i')^2$$ where $\Delta_i$ is the leading priciple minor.

such a basis are not unique

The proof is constructive, I will provide it later.

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  • $\begingroup$ Thanks @Irvingo, but be sure to read the full question before answering. It's clear what Jacobi's method is and how it works (and is proven) but the question I have is, what is the common terminology for this, as "Jacobi's Method" is only used in Russian/Soviet textbooks. $\endgroup$ Apr 25 '21 at 19:45

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