Standardize a Quadratic Form

Question

standardize Quadratic Form

$$8x_1x_4+2x_3x_4+2x_2x_3+8x_2x_4$$

how to do?

what's the simplest method.

How to choose the first linear replacement,

matrix or do by completing the square.

I've tried several times, but failed.

Answer 1

I am not sure what is the simplest solution method, but with three or four variables, it is usually easy to diagonalise a quadratic form by hand with blockwise Gaussian elimination and blockwise eigen-decomposition. We have \begin{align*} \pmatrix{1&0&-4&0\\ 0&1&-4&-1\\ 0&0&1&0\\ 0&0&0&1} \pmatrix{0&0&0&4\\ 0&0&1&4\\ 0&1&0&1\\ 4&4&1&0} \pmatrix{1&0&0&0\\ 0&1&0&0\\ -4&-4&1&0\\ 0&-1&0&1} &=\pmatrix{0&-4&0&0\\ -4&-8&0&0\\ 0&0&0&1\\ 0&0&1&0},\\ \pmatrix{1&\tfrac{-1}2&0&0\\ 0&1&0&0\\ 0&0&\tfrac{1}{\sqrt{2}}&\tfrac{1}{\sqrt{2}}\\ 0&0&\tfrac{-1}{\sqrt{2}}&\tfrac{1}{\sqrt{2}}} \pmatrix{0&-4&0&0\\ -4&-8&0&0\\ 0&0&0&1\\ 0&0&1&0} \pmatrix{1&0&0&0\\ \tfrac{-1}2&1&0&0\\ 0&0&\tfrac{1}{\sqrt{2}}&\tfrac{-1}{\sqrt{2}}\\ 0&0&\tfrac{1}{\sqrt{2}}&\tfrac{1}{\sqrt{2}}} &=\pmatrix{2&0&0&0\\ 0&-8&0&0\\ 0&0&1&0\\ 0&0&0&-1}, \end{align*} where blockwise Gaussian elimination is used on the first and second lines, and eigen-decomposition is performed on the bottom right $2\times 2$ subblock on the second line. Therefore, if \begin{align*} x =\pmatrix{1&0&0&0\\ 0&1&0&0\\ -4&-4&1&0\\ 0&-1&0&1} \pmatrix{1&0&0&0\\ \tfrac{-1}2&1&0&0\\ 0&0&\tfrac{1}{\sqrt{2}}&\tfrac{-1}{\sqrt{2}}\\ 0&0&\tfrac{1}{\sqrt{2}}&\tfrac{1}{\sqrt{2}}}y, \end{align*} then $$ x^\top\pmatrix{0&0&0&4\\ 0&0&1&4\\ 0&1&0&1\\ 4&4&1&0}x \ =\ y^\top\pmatrix{2&0&0&0\\ 0&-8&0&0\\ 0&0&1&0\\ 0&0&0&-1}y \ =\ 2y_1^2-8y_2^2+y_3^2-y_4^2. $$ To solve the problem more systematically, you may perform an orthogonal diagonalisation on the matrix representation of the quadratic form. However, this involves finding the eignvalues of a $3\times3$ or $4\times4$ matrix, which is not easy (and unnecessary) if you do it by hand.