Is a Zero Principal Minor in PSD Matrices Indicative of Smaller Zero Minors?

In summary, a symmetric Positive semidefinite (PSD) matrix is a square matrix where all the eigenvalues are non-negative. These matrices have various applications in mathematics and engineering, particularly in machine learning and data analysis. The condition on principle minors of a PSD matrix requires all principal minors to be non-negative, which is a weaker condition compared to positive definiteness. Therefore, a non-symmetric matrix cannot be PSD, but it can be positive semidefinite if its symmetric part satisfies the condition on principle minors.
  • #1
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Hi everyone,
Let [tex] A=(a_{ij})[/tex] be a symmetric (i.e., over reals) PSD matrix. Then is the following correct?

"If any principle minor ( [tex] \ne A [/tex] ) be zero, then all principle minor contained in this minor should also be zero".

I can not prove or disprove it..any help?

By the way how the result will change if we consider Hermitian matrix (over complex) instead of symmetric matrix?

Thanks
 
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  • #2
Oh..I got the answer. Its not correct. Consider the diagonal matrix: D={1,1,0,1,...}. Clearly $A_33=0$ but $A_22$ is non-zero.
 

FAQ: Is a Zero Principal Minor in PSD Matrices Indicative of Smaller Zero Minors?

1. What is a symmetric Positive semidefinite (PSD) matrix?

A symmetric Positive semidefinite (PSD) matrix is a square matrix where all the eigenvalues are non-negative. In other words, the matrix must be symmetric and all the eigenvalues must be greater than or equal to zero.

2. What is the significance of a PSD matrix?

PSD matrices have many important applications in mathematics and engineering. They are used in optimization problems, signal processing, and statistics. In particular, they are often used in machine learning and data analysis.

3. What is a condition on principle minors of a PSD matrix?

The condition on principle minors of a PSD matrix states that all the principal minors (determinants of the submatrices formed by selecting k rows and k columns) must be non-negative. This is a necessary condition for a matrix to be PSD.

4. How is the condition on principle minors of a PSD matrix related to positive definiteness?

If a matrix satisfies the condition on principle minors, then it is positive semidefinite (PSD). However, if all the principal minors are strictly positive, then the matrix is positive definite (PD). In other words, the condition on principle minors is a weaker condition compared to positive definiteness.

5. Can a non-symmetric matrix be PSD?

No, a non-symmetric matrix cannot be PSD. The matrix must be symmetric in order to satisfy the condition on principle minors. However, a non-symmetric matrix can be positive semidefinite if it satisfies the weaker condition on the principal minors of its symmetric part.

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