Quadratic Form $f\left(x_1,x_2,\cdots ,x_n\right)=\sum _{i=1}^m \left(a_{i 1}x_1+\cdots +a_{i n}x_n\right)^2$

Question

Quadratic Form $f\left(x_1,x_2,\cdots ,x_n\right)=\sum _{i=1}^m \left(a_{i 1}x_1+\cdots +a_{i n}x_n\right)^2$,

i) write the corresponding matrix;

ii) when $a_{\text{ij}}$ are all real numbers, gives the condition, when the quadratic form is positive.

• i)

\begin{align*}\left|\begin{array}{cccc} \sum _{i=1}^n a_{i 1}^2 & \sum _{i=1}^n a_{i 1}a_{i 2} & \cdots & \sum _{i=1}^n a_{i 1}a_{i n} \\ \sum _{i=1}^n a_{i 1}a_{i 2} & \sum _{i=1}^n a_{i 1}^2 & \cdots & \sum _{i=1}^n a_{i 2}a_{i n} \\ \cdots & \cdots & \cdots & \cdots \\ \sum _{i=1}^n a_{i 1}a_{i n} & \cdots & \cdots & \sum _{i=1}^n a_{i n}^2 \\\end{array}\right|\end{align*}

Is it right?

• ii)

How to do this?

Answer 1

Hint: Define $$ A=\left(\begin{array}{cccc} a_{11} & a_{12} &\cdots & a_{1n} \\ a_{21} & a_{22} &\cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} &\cdots & a_{nn} \\ \end{array}\right) $$ and $$ X^t=(x_1,x_2,\cdots,x_n) $$ Then $f=(A\cdot X)\cdot(A\cdot X)=X\cdot (A^tA\cdot X)=X^t\cdot(A^tA)\cdot X=||A\cdot X||^2$.