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Linear Algebra: Symmetric/Positive Definite problem

  • Thread starter Scootertaj
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  • #1
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1. Let A[itex]\in[/itex]Rnxn be a symmetric matrix, and assume that there exists a matrix B[itex]\in[/itex]Rmxn such that A=BTB.
a) Show that A is positive semidefinite
B) Show that if B has full rank, then A is positive definite




2. Homework Equations :
This is for an operations research class, so most of the definitions revolve around minimizing/maximizing.
However, alternate definitions:
Positive Definite: A is positive definite if for all non-null vectors h, hTAh > 0.
Symmetric: if AT=A.
Semidefinite: hTAh ≥ 0



The Attempt at a Solution


Here's some work:
AT = A ; A = BTB.
So, AT = BTB → ATA = BTBA = AA = A2.
So, ATA ≥ 0.
But, that's not quite what I want.
 
Last edited:

Answers and Replies

  • #2
Dick
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[itex]h^T A h=h^T B^T B h[/itex]. That's the inner product [itex](B h)^T (B h)[/itex]. Use the properties of the inner product.
 
Last edited:
  • #3
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Dick,
I don't recall any specific properties of inner products that would help except <x,x> >= 0. But, I don't see how that applies.
 
  • #4
Dick
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Dick,
I don't recall any specific properties of inner products that would help except <x,x> >= 0. But, I don't see how that applies.
And <x,x>=0 only if x=0. I think it applies a lot. [itex](B h)^T (B h)=<Bh, Bh>[/itex].
 
  • #5
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D'oh! I must be too tired, completely forgot that the inner product would be the same as doing the transpose first.
Thank you a lot Dick, you always seem to help out a lot.
 

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