B is nonsingular -- prove B(transpose)B is positive definite.

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SUMMARY

The discussion focuses on proving that if B is a real nonsingular matrix, then the matrix BTB is symmetric and positive definite. The proof of symmetry is established by demonstrating that the diagonal elements are equal. The positive definiteness is shown using the expression uTBTBu = (Bu)T(Bu), which confirms that the result is greater than zero for any non-zero vector u, without needing the nonsingularity condition of B.

PREREQUISITES
  • Understanding of matrix operations and properties
  • Familiarity with the concepts of symmetric and positive definite matrices
  • Knowledge of linear algebra, particularly regarding nonsingular matrices
  • Proficiency in using vector notation and inner products
NEXT STEPS
  • Study the properties of positive definite matrices in linear algebra
  • Learn about the implications of matrix nonsingularity on transformations
  • Explore the derivation and applications of the Gram-Schmidt process
  • Investigate the relationship between eigenvalues and positive definiteness
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Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, or related fields, will benefit from this discussion.

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Homework Statement


Suppose B is a real nonsingular matrix. Show that: (a) BtB is symmetric and (b) BtB is positive definite

2. Homework Equations
N/A

The Attempt at a Solution


I have managed to prove (a) by showing that elements that are symmetric on the diagonal are equal. However I have no idea how to prove B. I've tried to express [cij] with sigma notation with no sucess. I've also tried to apply the rules of matrices to try and show that utBtBu is bigger than zero with no sucess as well (perhaps (Bu)t=uB?...). Any help would be greatly appriciated.
 
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Well, u^TB^TBu = (Bu)^T(Bu) and you haven't yet needed the condition that B is nonsingular...
 
pasmith said:
Well, u^TB^TBu = (Bu)^T(Bu) and you haven't yet needed the condition that B is nonsingular...
Oh wow that rule completely slipped out of my mind. Thank you for your help.
 

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