Rank(AB) = Rank(A)Rank(B)?The Rank Product Theorem for Matrices

The determinant of a product of two matrices is equal to the product of their determinants. Rank(AB) = Rank(A)Rank(B)?Yes, this is true. The rank of the product of two matrices is equal to the product of their ranks.In summary, the product of two positive definite (real) matrices is positive definite, the determinant of a product of two matrices is equal to the product of their determinants, and the rank of the product of two matrices is equal to the product of their ranks.
  • #1
itpro
1
0
Can someone point me to the proof or give it here for the claim that product of the two positive definite (real) matrices is positive definite.

How about determinants of two matrices? Is det(AB) = det(BA)

Rank(AB) = Rank(A)Rank(B)?

Thank you in advance.

This is not a homework question. If you need a proof or a source for my claim please send PM.
 
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  • #2

FAQ: Rank(AB) = Rank(A)Rank(B)?The Rank Product Theorem for Matrices

1. What is a product of positive definite matrices?

A product of positive definite matrices refers to the multiplication of two or more matrices that are positive definite. A positive definite matrix is a square matrix where all of its eigenvalues are positive. This type of matrix is often used in statistics and engineering applications.

2. How do you determine if two matrices are positive definite?

To determine if a matrix is positive definite, you can check its eigenvalues. If all of the eigenvalues are positive, then the matrix is positive definite. Another way is to check if the matrix satisfies the positive definite property, which states that for any non-zero vector x, xTAx > 0, where A is the matrix in question.

3. What is the significance of positive definite matrices in statistics?

Positive definite matrices are used in statistics to represent covariance matrices, which are used to describe the variability and relationships between different variables. These matrices are also used in multivariate analysis and regression models.

4. Can a product of positive definite matrices result in a non-positive definite matrix?

No, a product of positive definite matrices will always result in a positive definite matrix. This is because the product of two positive definite matrices will have eigenvalues that are the product of the eigenvalues of the individual matrices, which will still be positive.

5. How are positive definite matrices used in engineering applications?

In engineering, positive definite matrices are used in optimization problems, such as in structural design and control systems. They are also used in solving differential equations and in finite element analysis. Positive definite matrices are also important in convex optimization, which is used in many engineering applications.

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