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Results tagged with matrix-analysis
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user 121643
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.
1
vote
Showing a 2-by-2 matrix is a contraction
Taking $B=\begin{pmatrix}0&\sqrt{2}\\0&0\end{pmatrix}$ and $S=\{z\in\mathbb{C}, |z|\le \frac{1}{\sqrt{2}}\}$ is a counterexample. The bound is $\sigma_1(B)\le \sqrt{2}$.
For any matrix $B,M\in \mathbb …
- 785
1
vote
An inequality related to matrix trace
The inequality fails when we take non particular cases, for example non diagonal matrices. Say $U \Sigma U^T>0$, and $S=(U \Sigma U^T)^{-1}$. The inequality is equivalent to prove $$\text{Tr}(SB^T|B|) …
- 785
0
votes
A (linear) optimization problem subject to (linear) matrix inequality constraints
Up to a unitary matrix $U$, $X$ is diagonal with all eigenvalues between $0$ and $1/2$ the other constraint on $AX$ implies that the diagonal entries of $U^*AUU^*XU$ are $\le 0$ and so are those of $ …
- 785
4
votes
Accepted
What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients?...
This may answer the question, the sequence is the only implicit thing. I consider $Q=\begin{pmatrix}Q_{i,j}\end{pmatrix}_{n\times n}$ for $n\ge 3$, where
$$Q_{i,j}=
\begin{cases}
\dbinom{j}{i-1}, & 1\ …
- 785
1
vote
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
A bound seems $4n(n-1)$, attained for $A=J-I$ and $B=-A$ (as in the comment) -edit- if we allow any ordering of the eigenvalues. First $$\sum_{i=1}^n\lambda_i(A)^2=\text{Tr}(A^2)=\sum_{i,j}|a_{i,j}|^2 …
- 785
1
vote
Extend an inequality on matrix norms
I post this instead of a comment, feel free to comment.
The inequality in the main question for $p=1$ (and $ p>1$) can be found in chap. 3 Topics in matrix analysis by R. Horn and C. Johnson, (i don …
- 785