Eigenvalues of positive definite (p.d) matrix

In summary, the conversation discusses the relationship between the eigenvalues of a positive definite matrix C and the eigenvalues of matrices A and B when C = A + B. The conversation mentions that real symmetric positive definite matrices have a geometrical interpretation as (hyper)ellipsoids in n-space, with eigenvectors representing the directions of the ellipsoid radii and eigenvalues representing their lengths. The suggestion is made to study how adding two positive definite matrices affects the geometry of the ellipsoids, potentially providing a visual understanding of the relationship. The speaker expresses interest in finding a solution to this problem and encourages others to share any insights they may have.
  • #1
roee
1
0
If C = A +B where A,B are both p.d, than C is p.d and its eigenvalues are positive.

Waht can you say about the relationship between the eigenvalues of C, and A,B ?

Thanks.
 
Physics news on Phys.org
  • #2
Hello,
There is for sure a (non-trivial) relationship but I cannot provide yet a concrete answer, because I myself am working on the same problem at the moment.

I can recall that real symmetric positive definite matrices have a geometrical interpretation: they represent (hyper)ellipsoids in n-space (check the relative lecture at MIT of Gilbert Strang here: http://www.mathvids.com/lesson/math...-tests-tests-for-minimum-and-ellipsoids-in-rn)
The eigenvectors of the matrix represent the directions of the ellipsoid radii, while the corresponding eigenvalues represent their lenghts.

I suspect that studying how addition of two PD matrices affects the geometry of the ellipsoids might give you at least a visual understanding and put you on the right track. I believe that such concepts are used in physics too, but physicists usually refer to 2nd-rank symmetric tensors.

Please if you find a solution to this problem post it here! This is an interesting problem. I also posted some similar questions here in PF, and (quite to my surprise) have never got any answer.
 
  • #3
something useful might be only if they share an eigenvector
 

1. What are eigenvalues of a positive definite matrix?

Eigenvalues of a positive definite matrix are the set of real numbers that, when multiplied by a corresponding eigenvector, result in the original vector multiplied by a positive scalar. In other words, they are the values that satisfy the equation Ax = λx, where A is the positive definite matrix, x is the eigenvector, and λ is the eigenvalue.

2. How do you find the eigenvalues of a positive definite matrix?

There are a few methods for finding the eigenvalues of a positive definite matrix, including the power iteration method and the Jacobi eigenvalue algorithm. These methods involve repeatedly applying matrix operations until the eigenvalues converge to their final values.

3. What is the relationship between the eigenvalues of a positive definite matrix and its diagonal entries?

The eigenvalues of a positive definite matrix are always greater than zero and are equal to the diagonal entries of the matrix. This means that the diagonal entries can be used to determine whether a matrix is positive definite.

4. Can a positive definite matrix have negative eigenvalues?

No, a positive definite matrix cannot have negative eigenvalues. By definition, a positive definite matrix must have all positive eigenvalues. If a matrix has even one negative eigenvalue, it is not positive definite.

5. What is the significance of eigenvalues in positive definite matrices?

Eigenvalues of a positive definite matrix are important because they provide information about the behavior of the matrix, such as its stability and convergence properties. They are also used in many applications, including optimization, differential equations, and statistical analysis.

Similar threads

I Getting eigenvalues of an arbitrary matrix with programming
    • Linear and Abstract Algebra
Replies
1
Views
805
I How can I test for positive semi-definiteness in matrices?
    • Linear and Abstract Algebra
Replies
1
Views
594
I Proof about a positive definite matrix
    • Linear and Abstract Algebra
Replies
12
Views
1K
A Eigenvalues of block matrix/Related non-linear eigenvalue problem
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Solving matrix commutator equations?
    • Linear and Abstract Algebra
Replies
6
Views
509
I If L is diagonalizable, algebraic & geometric multiplicities are equal
    • Linear and Abstract Algebra
Replies
4
Views
2K
I How do I know if a matrix is positive definite?
    • Linear and Abstract Algebra
Replies
11
Views
2K
I Markov Process: How to Tell Reversibility & Eigenvalues=1
    • Linear and Abstract Algebra
Replies
12
Views
1K
Determinant and symmetric positive definite matrix
    • Linear and Abstract Algebra
Replies
1
Views
2K
I Is a symmetric matrix with positive eigenvalues always real?
    • Linear and Abstract Algebra
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top