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Results tagged with matrices
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user 17261
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
7
votes
What are the possible eigenvalues of these matrices?
Consider the case where the $8\times 8$ matrix is positive semidefinite and assume that the 5 largest eigenvalues are all equal. Then by the argument of Federico Poloni they equal a. Then it follows $ …
- 2,753
4
votes
Accepted
What are the possible eigenvalues of these matrices?
Let $V$ be the real vector space of the $8\times 8$ matrices of the form given in the question.
Where is an open set in $\mathbb{R}^8$ of possible $8$-tuples of eigenvalues of matrices in $V$. … So I need $7$ additional matrices $M_2,...,M_8$ in $V$ such that the matrix
$X = (v_i^* M_j v_i)_{ij}$ is nonsingular .
Let $f(a,b,c,A,B)$ be the corresponding matrix. …
- 2,753
3
votes
Accepted
A generalized log inequality for positive definite trace-one matrices
Fedor's proof can be generalized :
Let $S = X^{1/2}$, $U_i = V_i S^{1/2}$ and $W_i = U_i (U_i^T S U_i)^{-1/2}$ .
Then $W_i^T S W_i = I_m$ where $I_m$ is the $m\times m$ identity .
Since $log(x) \ge …
- 2,753
3
votes
Accepted
Is $k(A, B) = \text{Tr}[(A^{1/2} B A^{1/2})^{1/2}]$ a positive definite kernel?
Counterexample for $n = 2$ :
Let $A_k$ be the orthonormal projection on the span of $$(\cos(2 \pi (k-1) / 5), \sin(2 \pi (k-1) / 5))^\mathsf{T} , \quad k = 1...5.$$
Then $k(A_k,A_l) = \vert \cos(2 \pi …
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3
votes
Accepted
A simple but curious determinantal inequality
Let $C = A^{-(k+1)/2} B A B A^{-(k+1)/2}$ and $D = A^{-(k-1)/2} B A^{-1} B A^{-(k-1)/2}$ .
We have to show that $det(I+C) \ge det(I+D)$ .
Now my goal is to apply equation (5.21) in Ando, Majorizatio …
- 2,753
2
votes
Bounding a spectral gap: what proof techniques exist?
For VBS quantum antiferromagnets in one dimension see also :
Ian Affleck, Tom Kennedy, Elliott H. Lieb and Hal Tasaki,
Valence bond ground states in isotropic quantum antiferromagnets.
Comm. Math. Ph …
5
votes
Accepted
How to prove this determinant is positive-II?
Let $q(x,y) = x^H J y$ for $x,y \in \mathbb{C}^{2n}$ where $J = diag(I_n,-I_n)$ and let $S = \{A \in M_{2n}(\mathbb{R}) : q(Ax, Ax) \ge q(x,x) $ $\forall x \in \mathbb{C}^{2n}\}$.
Obviously $S$ is a s …
- 2,753
18
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ ma...
Let $S$ be the span of $SO(N)$ .
Then it's obvious that if $A \in S$ and $D_1, D_2 \in SO(N)$ then $D_1^{-1} A D_2 \in S$ .
Therefore it's enough to show that $B := diag(1,0,...0) \in S$ .
If $N$ i …
- 2,753
6
votes
How to prove this determinant is positive?
The conjecture ist true :
It follows from $$det(I + e^{M}) = det(I + i e^{M/2}) * det(I - i e^{M/2}) = {\left\lvert{det(I + i e^{M/2})}\right\lvert}^2 \ge 0$$ .
I.e. it follows from the fact that ev …
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4
votes
Accepted
A singular value-eigenvalue inequality
The conjecture is true.
Lemma 1 : For every matrix $Z$ holds $\lambda_j(Z^* Z + Z^* + Z ) \leq \lambda_j(Z^* Z + 2 (Z^* Z)^{1/2})$ .
For a proof see the proof of $\lambda_j(Z^* + Z ) \leq \lambda_j …
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3
votes
Accepted
An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices
The answer is no if there exists $z \in S$ with $z \neq 1$ and $\vert 1 - z\vert \leq \frac{1}{2}$ and $\bar{z} \in S$ :
Let $$p(x) = \sum_{k=0}^{n-1} b_k x^k$$ and $x_k = p(w^k)$ , where $w=e^{\frac …
- 2,753