Another necessary condition for Positive Semidefiniteness?

In summary, the conversation discusses a rule or theorem about hermitian Positive Semidefinite matrices, stating that the largest entry (considering modulus) must be on a diagonal. This can sometimes be a tricky step in proving that a matrix is not PSD. The Schur complement formula is suggested as a resource for further understanding.
  • #1
NaturePaper
70
0
Hi everyone in this sub forum,
I'm wondering if the following 'rule' (theorem?) is correct:
For a hermitian Positive Semidefinite (PSD) matrix [tex]A=(a_{ij})[/tex],
[tex]\max_{i,j\le n} |a_{ij}|=\max_{i\le n}a_{ii}[/tex].


The reason for this intuition (It may be a well known result, I'm very sorry in this
case for my poor knowledge) is the following:

A is PSD [tex]\Rightarrow[/tex] all its [tex]2\times2[/tex] Principal submatrices are PSD
[tex]\Rightarrow~~\left[\begin{array}{cc}
a_{ii} & a_{ij} \\
\bar{a}_{ij} & a_{jj} \end{array}\right]\ge0
[/tex]
[tex]\Rightarrow~~~|a_{ij}|\le \sqrt{a_{ii}a_{jj}}[/tex]
[tex]\Rightarrow~~\max_{i,j\le n} |a_{ij}|=\max_{i\le n}a_{ii}[/tex].

Regards,
NaturePaper
 
Last edited:
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  • #2
Since I get no reply, I think I'd rather state what it means:

"For a PSD matrix the largest (consider modulus) entry should necessarily be on a diagonal".

This sometimes may be a tricky step to prove that a matrix is not PSD.

Is it write?

Regards
 
Last edited:
  • #3
Try Schur complement formula to get a feeling for all these issues...
 
  • #4
@trambolin,
Please let me know whether what I said (guessed) is wrong. Where is the discrepancy?

Regards,
NP
 
  • #6
@trambolin,
Thanks. Its correct.
 

Related to Another necessary condition for Positive Semidefiniteness?

1. What is Positive Semidefiniteness?

Positive Semidefiniteness is a mathematical concept that is often used in linear algebra and optimization problems. It refers to a property of a matrix where all of its eigenvalues are non-negative. In other words, the matrix contains only non-negative values and has a positive or zero determinant.

2. Why is Another Necessary Condition for Positive Semidefiniteness important?

Another necessary condition for Positive Semidefiniteness is important because it helps us determine whether a given matrix is positive semidefinite or not. This condition states that all principal submatrices of a positive semidefinite matrix must also be positive semidefinite. This condition is useful in proving the positive semidefiniteness of a matrix and also in applications such as convex optimization problems.

3. How is Another Necessary Condition for Positive Semidefiniteness related to the Positive Semidefinite Property?

The Positive Semidefinite Property is a broader concept that encompasses the Another Necessary Condition for Positive Semidefiniteness. The positive semidefinite property states that a matrix must have all non-negative eigenvalues, while the other necessary condition specifies that all principal submatrices must also have this property. Therefore, this condition is a subset of the positive semidefinite property.

4. Can a matrix satisfy the Another Necessary Condition for Positive Semidefiniteness but not be positive semidefinite?

No, if a matrix satisfies the Another Necessary Condition for Positive Semidefiniteness, then it must also be positive semidefinite. This is because all principal submatrices of a positive semidefinite matrix are themselves positive semidefinite. However, a matrix can be positive semidefinite without satisfying this condition if it has other properties that guarantee its positive semidefiniteness.

5. How is Another Necessary Condition for Positive Semidefiniteness used in real-world applications?

Another Necessary Condition for Positive Semidefiniteness is used in various applications such as in the field of machine learning, where it is used to ensure that optimization problems are well-defined and have a unique solution. It is also used in engineering and physics to determine the stability and positive definiteness of systems. Additionally, this condition is useful in the analysis of financial data and in statistics to ensure that covariance matrices are positive semidefinite.

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