5
$\begingroup$

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by considering rank of $A$ and $[A|b]$.

If we consider the equation $x^tAx=\lambda$, for $\lambda \in K$, what are the criteria for the existance of solution and simple methods to solve it?

  • 0
    Regarding $x^t A x = \lambda$ when $A$ is positive (negative) semi-definite, then the equation has no solutions for \lambda<0 (\lambda>0). In the more general case, if $A$ is diagonalizable then you can write it in the form $A=V\Lambda V^{-1}$ where $\Lambda$ is diagonal and you equation becomes: \begin{equation} x^{t}\Lambda$x$= V^{-1}\lambda V =: \xi\end{equation}2011-08-22

0 Answers 0