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Results tagged with matrix-analysis
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user 1131
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
3
votes
Accepted
Upper bound of the largest eigenvalue of a PSD block matrix in terms of blocks
Write $A=\sum w_i\otimes w_i$ where $w_i$ are orthogonal and $\|w_1\|=\max_i\|w_i\|$. Also write $w_i=(u_i,v_i)$. Then the inequality in question is just
$$
\|u_1\|^2+\|v_1\|^2+\max_{e:\|e\|=1}\sum_i …
- 61.9k
8
votes
Accepted
Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$
Just use Cauchy-Schwarz and the identity
$$
\sum_{i,j}(u_iu_j-v_iv_j)^2=\left[\sum_i u_i^2\right]^2+\left[\sum_i v_i^2\right]^2-2\left[\sum_i u_iv_i\right]^2=2
$$
to get $f(u,v)\le \sqrt2 n$ for all $ …
- 61.9k
5
votes
Accepted
Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?
This Asymptote code seems to work perfectly and for any $t$ in your range the estimate uses $Cd$ operations and is a guaranteed upper bound though I am not sure whether $C$ is small enough for you (I …
- 61.9k
7
votes
Accepted
Bound the eigenvalue of product of matrices?
OK, looks like I've got it. Since it is pretty late here now, it would be nice if someone could check that I haven't written some nonsense (I apologize in advance if I have).
Let $N(H)$ be the kernel …
- 61.9k
8
votes
Operator norms of circulant matrices
The answer to Question 2 is "No". Note that the $2$-norm is just the maximum of $\left|\sum_{k=0}^{n-1}a_{k+1}z^k\right|$ over the $n$-th power roots of unity and the $1$-norm is just $\sum_k |a_k|$. …
- 61.9k
5
votes
Accepted
Bounding the non-multiplicativity of isometric projection
If the constant is allowed to depend upon the dimension, the estimate is simple. Let $A=O_AP_A,B=O_BP_B$, then $AB=O_AO_B(O_B^*P_AO_B)P_B$ and we are left with showing that if the product of two posit …
- 61.9k
5
votes
Accepted
Trace of a nonlinear matrix equation (cont'd)
Actually, you have completely solved it yourself, just didn't dare to acknowledge it. In my notation, you have $(X\circ X^T)v(A)=(Y\circ Y^T)v(I)$ when $Y^2=XAX$. Similarly, $(Z\circ Z^T)v(I)=(Y\circ …
- 61.9k
5
votes
Accepted
On a matrix inequality
I'll try to answer both questions at once. First, let's find the reason why the LHS is positive at all. I claim that the spectrum $\mu_1\le\mu_2\le\dots\le\mu_n$ of $2\sqrt{A^{1/2}BA^{1/2}}$ is domina …
- 61.9k
6
votes
Accepted
Does approximate equality of quantum states imply operator inequality in a large subspace?
Let $\sigma$ be represented by a PD matrix $A$ and $\rho$ by $A+B$. Note that $|B|\ge B$ (in the sense of PSD matrices) and has the same $1$-norm. Also $\Pi |B|\Pi\ge \Pi B \Pi$, so to dominate $\Pi B …
- 61.9k