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

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SUMMARY

The discussion centers on the properties of positive definite matrices, specifically addressing the claim that the product of two positive definite matrices is also positive definite. It references the Hadamard product, confirming that the Hadamard product of two positive (semi)definite matrices remains positive (semi)definite. Additionally, the discussion touches on the determinant property, questioning whether det(AB) equals det(BA), which is established as true for square matrices.

PREREQUISITES
  • Understanding of positive definite matrices
  • Familiarity with matrix multiplication
  • Knowledge of determinants in linear algebra
  • Concept of the Hadamard product
NEXT STEPS
  • Research the proof of the positive definiteness of matrix products
  • Explore the properties of the Hadamard product in detail
  • Study the implications of determinant properties in linear algebra
  • Investigate applications of positive definite matrices in optimization problems
USEFUL FOR

Mathematicians, students of linear algebra, and professionals working with matrix theory or optimization will benefit from this discussion.

itpro
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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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