Let
be a set of orthonormal vectors with
, 2, ...,
, such that the inner product
.
Then set
![x=sum_(k=1)^Ku_ky^k](/images/equations/PoincareSeparationTheorem/NumberedEquation1.svg) |
(1)
|
so that for any square matrix
for which the product
is defined, the corresponding quadratic
form is
![(x,Ax)=sum_(k,l=1)^Ku_ku_l(y^k,Ay^l).](/images/equations/PoincareSeparationTheorem/NumberedEquation2.svg) |
(2)
|
Then if
![B_k=(y^k,Ay^l)](/images/equations/PoincareSeparationTheorem/NumberedEquation3.svg) |
(3)
|
for
,
2, ...,
,
it follows that
![lambda_i(B_K)<=lambda_1(A)](/images/equations/PoincareSeparationTheorem/NumberedEquation4.svg) |
(4)
|
![lambda_(K-j)(B_K)>=lambda_(N-j)(A)](/images/equations/PoincareSeparationTheorem/NumberedEquation5.svg) |
(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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