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

In summary, the conversation is about proving that BtB is both symmetric and positive definite, given that B is a real nonsingular matrix. The first part of the proof involves showing that elements on the diagonal of BtB are equal, while the second part involves using the rule u^TB^TBu = (Bu)^T(Bu) to show that BtB is positive definite. The condition of B being nonsingular is not required for the proof.
  • #1
Adgorn
130
18

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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  • #2
Well, [itex]u^TB^TBu = (Bu)^T(Bu)[/itex] and you haven't yet needed the condition that [itex]B[/itex] is nonsingular...
 
  • #3
pasmith said:
Well, [itex]u^TB^TBu = (Bu)^T(Bu)[/itex] and you haven't yet needed the condition that [itex]B[/itex] is nonsingular...
Oh wow that rule completely slipped out of my mind. Thank you for your help.
 

What does it mean for B to be nonsingular?

For a matrix B to be nonsingular, it means that it has a determinant that is not equal to zero. This implies that B has an inverse matrix B-1 that can be used to solve linear equations involving B.

What does it mean for BTB to be positive definite?

A matrix A is positive definite if all its eigenvalues are positive. For the product BTB to be positive definite, it means that all the eigenvalues of BTB are positive. In other words, BTB is a symmetric and positive definite matrix.

Why is it important to prove that BTB is positive definite?

Proving that BTB is positive definite is important because it guarantees that the matrix BTB is invertible. This has important implications in many areas of mathematics, including linear algebra, optimization, and statistics.

How can we prove that BTB is positive definite?

One way to prove that BTB is positive definite is to show that all its eigenvalues are positive. This can be done by using the Cholesky decomposition method or the Sylvester's criterion. Another approach is to show that BTB is a symmetric and positive semidefinite matrix, and then use the fact that positive semidefinite matrices have non-negative eigenvalues.

What are some practical applications of proving that BTB is positive definite?

Proving that BTB is positive definite has practical applications in many fields such as data analysis, machine learning, and signal processing. In data analysis, it can be used to determine the correlation between variables. In machine learning, it is used in the optimization of algorithms. In signal processing, it can be used to analyze and filter signals.

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