Condition on minors to be Positive Semidefinite (PSD)

In summary, it is necessary that all minors of order>=2 be non-negative for a hermitian matrix to be PSD.
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Hi everyone,

I know that for a hermitian matrix to be PSD it is necessary that every principal minor [i.e, the minors obtained by deleting all the last i rows and columns for all i=(n-1)(-1)0].

I want to know if it is necessary that all minors of order>=2 be non-negative.

Particularly, for the 4x4 hermitian matrix

a_11 a_12 a_13 a_14
a_21 a_22 a_23 a_24 (please read it as a matrix)
a_31 a_32 a_33 a_34
a_41 a_42 a_43 a_44

is it necessary that the minors
a_11 a_13
a_31 a_33

and

a_22 a_24
a_42 a_44

should be non-negative?

Please help.
 
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Related to Condition on minors to be Positive Semidefinite (PSD)

1. What does it mean for a condition to be positive semidefinite (PSD)?

A condition is positive semidefinite if its value is greater than or equal to zero for all possible inputs. In other words, it is a mathematical property that ensures the condition will always have a non-negative outcome.

2. Why is it important for minors to be PSD?

Minors being PSD is important because it guarantees that the matrix they are derived from is also positive semidefinite. This is a desirable property in many applications, such as optimization and control theory, as it ensures stability and convergence of algorithms.

3. How can I determine if a minor is PSD?

A minor can be determined to be PSD by checking if all of its eigenvalues are non-negative. If all eigenvalues are greater than or equal to zero, then the minor is positive semidefinite. Additionally, there are algorithms and software programs that can calculate the eigenvalues and determine if a minor is PSD.

4. What are some real-world applications of minors being PSD?

Minors being PSD have various applications in fields such as engineering, economics, and physics. For example, in control theory, positive semidefinite conditions are used to design stable and robust control systems. In economics, PSD conditions are used to ensure stable and efficient market equilibria. In physics, they are used to describe the stability of physical systems.

5. Can minors be PSD for all types of matrices?

Yes, minors can be PSD for all types of matrices, including square matrices, rectangular matrices, and even complex matrices. However, the specific conditions for PSD may vary depending on the type of matrix. For example, for a symmetric matrix, the condition is that all principal minors (minors formed by removing rows and columns from the top left corner) are non-negative.

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