A problem with matrix

[anaantonia_]

I need to write:

[math]x^2 + y^2 + 2z^2 +2xz - 2yz = 1[/math]
in matrix form:

[math]X^t AX =1[/math]
A, is a square matrix 3x3
X, is a vector

[math]X^t = \begin{pmatrix} x & y & z \end{pmatrix}[/math]
Can y'all help me?
?

 

[anaantonia_]

I found:

a(1.1) = 1
a(2.2) = 1
a(3.3) = 2
and a(2.1)= -a(1.2)

I need to discover the other values

 

[topsquark]

Expand it out!
[imath]\left ( \begin{matrix} x & y & z \end{matrix} \right ) \left ( \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right ) \left ( \begin{matrix} x \\ y \\ z \end{matrix} \right )[/imath]

I'll start:
[imath]= \left ( \begin{matrix} x & y & z \end{matrix} \right ) \left ( \begin{matrix} a_{11}x + a_{12}y + a_{13}z \\ a_{21}x + a_{22}y + a_{23}z \\ a_{31}x + a_{32}y + a_{33}z \end{matrix} \right )[/imath]

Multiply out the rest and you will have an expression that says [imath]a_{11} x^2 + a_{12} xy + a_{13} xz + \text{ ...}[/imath] and you can compare your expression to that to get the [imath]a_{ij}[/imath].

-Dan

 

[anaantonia_]

Hi topsquark, Thank you for your answer!

Expand it out!
[imath]\left ( \begin{matrix} x & y & z \end{matrix} \right ) \left ( \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right ) \left ( \begin{matrix} x \\ y \\ z \end{matrix} \right )[/imath]

I'll start:
[imath]= \left ( \begin{matrix} x & y & z \end{matrix} \right ) \left ( \begin{matrix} a_{11}x + a_{12}y + a_{13}z \\ a_{21}x + a_{22}y + a_{23}z \\ a_{31}x + a_{32}y + a_{33}z \end{matrix} \right )[/imath]

Multiply out the rest and you will have an expression that says [imath]a_{11} x^2 + a_{12} xy + a_{13} xz + \text{ ...}[/imath] and you can compare your expression to that to get the [imath]a_{ij}[/imath].

-Dan

So, I did and I found

[math]\begin{matrix} 1 & a1.2 & a1.3\\ a2.1 & 1 &a2.3\\ a3.1 & a3.2 & 2 \end{matrix}[/math]
and a2.1 = -a1.2.

There's another expression to associate or matrix properties that I have to considerate?

 

[topsquark]

Hi topsquark, Thank you for your answer!



So, I did and I found

[math]\begin{matrix} 1 & a1.2 & a1.3\\ a2.1 & 1 &a2.3\\ a3.1 & a3.2 & 2 \end{matrix}[/math]
and a2.1 = -a1.2.

There's another expression to associate or matrix properties that I have to considerate?

Oh! I see the problem. Barring any given reason not to I'd define A to be a symmetric matrix. That would make [imath]a_{12} = a_{21} = 0[/imath].

-Dan

 

[anaantonia_]

Oh! I see the problem. Barring any given reason not to I'd define A to be a symmetric matrix. That would make [imath]a_{12} = a_{21} = 0[/imath].

-Dan

It was... like magic. I got the answer. Thank youuuuuuuuuuuuuuuuu!
What led you to define it as a symmetric matrix? What the expression reveals for that analyses?
?