Can A>B Be Determined by A-B>0 for Matrices?

  • Context: Graduate 
  • Thread starter Thread starter matqkks
  • Start date Start date
  • Tags Tags
    Matrices
Click For Summary
SUMMARY

The discussion confirms that matrices can be compared using the notation A > B, which is valid when A - B is positive definite. This establishes a strict partial ordering for N x N matrices. Additionally, it is noted that if A > B and B > C, then A - C is also positive definite, demonstrating the transitivity of this ordering. The discussion emphasizes that both positive definite and positive semidefinite conditions are crucial for these comparisons.

PREREQUISITES
  • Understanding of positive definite and positive semidefinite matrices
  • Familiarity with matrix notation and operations
  • Knowledge of strict partial ordering concepts
  • Basic linear algebra principles
NEXT STEPS
  • Research the properties of positive definite matrices in linear algebra
  • Learn about the implications of matrix ordering in optimization problems
  • Explore the concept of semidefinite programming
  • Study the transitivity of relations in mathematical structures
USEFUL FOR

Mathematicians, data scientists, and anyone involved in linear algebra or optimization who needs to understand matrix comparisons and their implications in various applications.

matqkks
Messages
283
Reaction score
6
Can we compare matrices?
If A-B>0 is positive definite, can we say A>B?
 
Physics news on Phys.org
matqkks said:
Can we compare matrices?
If A-B>0 is positive definite, can we say A>B?
Yes, this is valid notation. This definition of ##>## gives us a strict partial ordering on the set of ##N\times N## matrices. Similarly, you can define ##A \geq B## if ##A - B## is positive semidefinite.

Note that if ##A > B## and ##B > C##, then ##A - B## and ##B - C## are positive definite, and ##A - C = (A - B) + (B - C)##. As the sum of two positive definite matrices is positive definite, this shows that the transitivity axiom of (partial or strict) ordering is satisfied. The other axioms are even easier to check.
 

Similar threads

Graduate Finding Eigenvectors for Two Matrices using the Generalized Jacobi Method
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Multinomial functions of matrices
  • · Replies 3 ·
Replies
3
Views
2K
MHB Does Commutativity Hold for Matrices A and B with a Specific Matrix C?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Matrix Multiplication: Solving A*A'=B for Unique A(B)
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Why are determinants in 2x2 matrices and 3x3 matrices computed the way they are?
  • · Replies 13 ·
Replies
13
Views
4K
Graduate Infinite matrices and the Trace function
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Solve a nonlinear matrix equation
  • · Replies 12 ·
Replies
12
Views
2K
MHB Set is closed as for multiplication of matrices
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad What is the proper matrix product?
  • · Replies 9 ·
Replies
9
Views
3K