We know $(\mathbf{x}-\mathbf{v})^TA(\mathbf{x}-\mathbf{v})\le1$ is a general ellipsoid.
I am doing a statistical calculation.
I wonder if $(\mathbf{x}-\mathbf{v})^TA(\mathbf{x}-\mathbf{v})\le d$ is still a ellipsoid?
Thanks
Is this still a Ellipsoid
0
$\begingroup$
linear-algebra
statistics
-
0Please explain (x-v) – 2017-02-14
-
0@Narasimham That is just translating the origin from $0$ to $v$. The $d$ influences the extension. First form looks like hull and interior, second form might give single point if $d = 0$ or outside if $d < 0$, assuming $A$ is positive definitite. – 2017-02-14
-
0Tag convex analysis? – 2017-02-14
1 Answers
2
If you replace $A$ in the first inequality by $d^{-1}A$, you get the second inequality. So yes, except your initial statement is incorrect: It is only true if $A$ is positive definite.
-
0Thanks, I see, A should be positive definite. – 2017-02-14