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Results tagged with matrix-analysis
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user 11260
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.
35
votes
Accepted
Consequences of eigenvector-eigenvalue formula found by studying neutrinos
The OP asks about generalisations and applications of the formula in arXiv:1908.03795.
$\bullet$ Concerning generalisations: I have found an older paper, from 1993, where it seems that the same resul …
- 188k
15
votes
Accepted
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
The name of an $n\times n$ matrix with positive real elements satisfying $a_{ij}=1/a_{ji}$ for all $i,j\in\{1,2,\ldots n\}$ is reciprocal matrix.
A consistent reciprocal matrix has elements of the for …
- 188k
11
votes
Matrix trace & norm
A proof of
$$\lambda_n(B)\,{\rm tr}\,A\leq {\rm tr}\,(AB)\leq\lambda_1(B)\,{\rm tr}\,A$$
where $\lambda_n$ is the $n$-th largest eigenvalue of $B$ so $||B||=\lambda_1(B)$ and $A$, $B$ are positive sem …
- 188k
8
votes
Accepted
An extension of the Golden-Thompson inequality
The difficulties with generalizations of the Golden-Thompson inequality to three matrices arise because the trace of a product of three positive symmetric matrices is in general not positive; unlike t …
- 188k
7
votes
Accepted
(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{...
Let me answer the first question:
Q: Does $F=\sum_{i=0}^{\infty} A^i (A^i)^{\top}$ for non-symmetric $A$ have a closed-form solution analogous to the solution $\sum_{i=0}^{\infty} A^{2i}=\left( I-A^2 …
- 188k
6
votes
Computation to differentiate a determinant
The condition $\lambda+\mu_1>0$ ensures that $M(\lambda)=A+\lambda $ is invertible, and then one can use Jacobi's formula
$$\frac{d}{d\lambda} \det M(\lambda) = \det M(\lambda) \operatorname{tr} \left …
- 188k
6
votes
Accepted
First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix
First order perturbation theory gives you
$$\frac{1}{\lambda_{\rm min}(A+\epsilon B)}=\frac{1}{\lambda_{\rm min}(A)}-\frac{\epsilon}{\lambda_{\rm min}(A)^2}\langle v_0|B|v_0\rangle+{\cal O}(\epsilon^2 …
- 188k
5
votes
Matrix integrals in combinatorics, for dummies
I'm not sure this qualifies as a "cookbook", but it's a tutorial introduction on the application of matrix integrals to combinatorics, with a quite complete overview of applications in a physics conte …
- 188k
5
votes
Taking matrix derivative with MATLAB or Wolfram Alpha
I presume by "taking the derivative" you mean that you want to know how this expression $\Omega(X)=a^TM(X)a$, with $M(X)=(I+\alpha X^TA^TAX)^{-1}$, changes when you change $X$ by a small amount $\delt …
- 188k
5
votes
Accepted
Matrix equation with projection matrix
The solution for $P$ to
$$P - G_1G_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0$$
is
$$P=(1 +\lambda )G_1 G_2 A^{-1},$$
as one can check by substitution into
$$G_1G_2P^\top(PAP^\top)^{-1}P=G_1 G_2( …
- 188k
4
votes
Inverse of a small submatrix
Let me denote $B=A^{-1}$. The question is how to efficiently compute the inverse of a submatrix of $B$ given the fact that the inverse of the full matrix $B$ is known (since $B^{-1}=A$). An efficient …
- 188k
3
votes
Fast Upper Triangular Matrix Exponentiation
The exponential $e^{Q}$ of any $n\times n$ upper triangular matrix $Q$ can be computed efficiently by solving a set of $n$ first-order differential equations, $u_{i}'(t)=\sum_{j}Q_{ij}u_j(t)$; these $ …
- 188k
3
votes
Accepted
$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?
$$A=\left(
\begin{array}{cc}
1 & \frac{1}{10} \\
\frac{1}{10} & 1 \\
\end{array}
\right),\;\;A^{-1}=\left(
\begin{array}{cc}
\frac{100}{99} & -\frac{10}{99} \\
-\frac{10}{99} & \frac{100}{99} \\
\ …
- 188k
3
votes
State-dependent positive definite matrix
Yes, it is possible; pick a nonzero element of $\mathbf{x}$, which must exist since $\mathbf{x}\neq 0$; let's say this nonzero element is $x_j$. Then define
$$\big(\mathbf{M}_{\mathbf{x}}\bigr)_{nm}=- …
- 188k
3
votes
Accepted
Question on integral expression of positive definite matrices
The formula
$$\frac{d}{ds}\log Z(s) = \int_0^1 [(1-t)I+tZ(s)]^{-1}Z'(s) [(1-t)I+tZ(s)]^{-1}\, dt,$$
with $Z(s)=X+sY$, gives upon integration of
$$\int_0^1 \frac{d}{ds}\log Z(s)\,ds=\log Z(1)-\log Z(0 …
- 188k