Constraint on M to keep M^T * A * M positive semidefinite?

In summary, the conversation discusses the condition for the inequality M^TAM\succeq 0 to hold, with the conclusion that it is always true for any M. The participants also mention that A is a given matrix and M is a variable.
  • #1
perplexabot
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Hey all! Let me get right to it!
It is given that $$A \succeq 0$$
I need the following to hold for [itex]M[/itex]: $$M^TAM\succeq 0$$

What are the constraints or conditions on [itex]M[/itex] for [itex]M^TAM\succeq 0[/itex] to hold?

Anything would help at this point... I am open to discussion.

Note: It may be worth mentioning that [itex]A[/itex] is a given matrix where as [itex]M[/itex] is variable.

Thank you for reading : )
 
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  • #2
It is always true. For any ##M##.
 
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  • #3
micromass said:
It is always true. For any ##M##.
Interesting, I think I see why that is. [itex]xM^TAMx \succeq 0 => y^TAy \succeq 0[/itex]

Thanks for the quick answer. I may have a follow up question later :P
 

1. What is a positive semidefinite matrix?

A positive semidefinite matrix is a square matrix where all of its eigenvalues are non-negative. This means that when multiplied by any non-zero vector, the resulting product will always be greater than or equal to 0.

2. What does the constraint on M to keep M^T * A * M positive semidefinite mean?

This means that in order for the resulting matrix M^T * A * M to be positive semidefinite, the matrix M must have certain properties or values that satisfy this condition. In other words, there are restrictions on the values that M can take in order to maintain the positive semidefiniteness of the resulting matrix.

3. Why is it important to keep M^T * A * M positive semidefinite?

Positive semidefinite matrices have many applications in mathematics and science, particularly in optimization and control theory. Keeping M^T * A * M positive semidefinite ensures that the resulting matrix can be used in these applications and will have certain desirable properties.

4. What are the consequences of not satisfying the constraint on M to keep M^T * A * M positive semidefinite?

If the constraint is not satisfied, the resulting matrix M^T * A * M may not be positive semidefinite. This could lead to incorrect results or may render the matrix unusable for certain applications where positive semidefiniteness is necessary.

5. How can the constraint on M to keep M^T * A * M positive semidefinite be satisfied?

The specific constraints on M will depend on the matrix A and the desired properties of M^T * A * M. In general, it may involve finding specific values or ranges of values for the elements of M that will result in a positive semidefinite matrix. This can be done through techniques such as eigenvalue analysis or optimization methods.

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