All Questions
7 questions
91
votes
5
answers
124k
views
Eigenvalues of matrix sums
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am ...
- 1,013
2
votes
1
answer
361
views
How the eigenvalues change when a Hermitian matrix is left multiplied and right multiplied by a diagonal matrix?
Suppose there is a Hermitian matrix $S$ with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_K$. There is a diagonal matrix $D$ whose entries on the main diagonal are positive. What are the ...
- 21
2
votes
0
answers
121
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
2
votes
0
answers
330
views
Eigenvalues of special sum of Hermitian matrices
In my research on linear algebra and its applications, I have come across the following problem which has stumped me:
Let $ A $ be a positive definite matrix and let $ D $ be a positive diagonal ...
- 620
1
vote
1
answer
6k
views
Largest eigenvalue of the sum of Hermitian matrices [closed]
Is there an expression for the largest eigenvalue of the sum of two Hermitian matrices in terms of the spectrum of the same matrices?
- 2,099
1
vote
1
answer
506
views
Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues
For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset
$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$
is a manifold of dimension $2n(2r)-(...
- 143
-3
votes
1
answer
270
views
Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)
For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
- 101