I am solving some of the problems given in Gilbert Strag's Introduction to Linear Algebra book in chapter 6. I came across one of the problems which couldn't solve. So I thought I should ask you guys how you would go for solving it. Below I post a picture taken of the problem taken from the book.
Could you please tell me how would you fill the entries in the matrix A?
Thank you
On
Expand the right-hand side to get:
$$ 4 x_1^2-8 x_2 x_1+16 x_3 x_1+4 x_2^2+16 x_3^2-16 x_2 x_3 $$
If we take $A$ to be symmetric, it is well known that the entry $a_{ij} = a_{ji} $ of the matrix $A$ is equal to half the coefficient of $x_i x_j$ if $i \neq j$ and the coefficient of $x_i^2$ otherwise. Namely:
$$ A = \begin{bmatrix} 4 & -4 & 8 \\ -4 & 4 & -8 \\ 8 & -8 & 16 \end{bmatrix} $$
I rewrite your problem with a bit less indices in the form: $$ \begin{bmatrix} x&y&z \end{bmatrix} \begin{bmatrix} a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&c_2&c_3\\ \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix}=4(x-y+2z)^2 $$ calculating the left side you find: $$ x(a_1x+a_2y+a_3z)+y(b_1x+b_2y+b_3z)+z(c_1x+c_2y+c_3z)=4(x-y+2z)^2 $$ So you have two plynomials in $x,y,z$ that have to be equals for all $(x,y,z)$ and this is true only if all the coefficents of the monomials are the same, so you can write a system of equations and find the unknowns $a_i,b_i,c_i$.