Let 
be a set of orthonormal vectors with 
, 2, ..., 
, such that the inner product 
.
Then set
 |
(1)
|
so that for any square matrix 
for which the product 
is defined, the corresponding quadratic
form is
 |
(2)
|
Then if
 |
(3)
|
for 
,
2, ..., 
,
it follows that
 |
(4)
|
 |
(5)
|
for 
,
2, ..., 
and 
,
1, ..., 
.
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References
Bellman, R. E. Introduction to Matrix Analysis, 2nd ed. New York: McGraw-Hill, p. 117, 1970.Gradshteyn,
I. S. and Ryzhik, I. M. Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
p. 1120, 2000.Referenced on Wolfram|Alpha
Poincaré Separation
Theorem
Cite this as:
Weisstein, Eric W. "Poincaré Separation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoincareSeparationTheorem.html
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