A condition on principle minors of a symmetric Positive semidefinite (PSD) matrix

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
NaturePaper
70
0
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
 
Physics news on Phys.org
  • #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.
 

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.

Similar threads

I Proof about a positive definite matrix
    • Linear and Abstract Algebra
Replies
12
Views
1K
I Problem with notation of matrix elements
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Is a symmetric matrix with positive eigenvalues always real?
    • Linear and Abstract Algebra
Replies
8
Views
2K
Condition on Leading principle minors of a symmetric Positive semidefinite(PSD) matri
    • Linear and Abstract Algebra
Replies
1
Views
3K
I How do I know if a matrix is positive definite?
    • Linear and Abstract Algebra
Replies
11
Views
2K
Semi-Positive Definiteness of Product of Symmetric Matrices
    • Linear and Abstract Algebra
Replies
4
Views
9K
Condition for a matrix to be Positive semi-definite
    • Linear and Abstract Algebra
Replies
17
Views
18K
Condition on minors to be Positive Semidefinite (PSD)
    • Linear and Abstract Algebra
Replies
1
Views
3K
Another necessary condition for Positive Semidefiniteness?
    • Linear and Abstract Algebra
Replies
5
Views
3K
MHB Zero-Trace Symmetric Matrix is Orthogonally Similar to A Zero-Diagonal Matrix.
    • Linear and Abstract Algebra
Replies
5
Views
5K

Hot Threads

Recent Insights

Back
Top