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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.
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
$$\int_0^1 \frac{d}{ds}\log Z(s)\,ds=\log Z(1)-\log Z(0 …
- 188k
1
vote
Accepted
The width of the minimum gap of an interpolated matrix
A bound with a function $f(G)$ that does not depend on $H(0)$ and $H(1)$ seems unlikely. In a typical situation the function $\gamma(s)$ can be extended to the complex plane and $\gamma(z)$ vanishes a …
- 188k
0
votes
Accepted
Gradient Descent for Markov Dynamics
Note that $A$ only appears in the combination $M=A-BC$, so the derivative with respect to $A$ equals the derivative with respect to $M$; The function $f(A)$ is given by
$$f(A)=||(A - BC)^Nv - w||_2^2= …
- 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
1
vote
How to estimate the norm of a matrix
You did not specify which norm, let me take the Frobenius norm, $||A||_F^2=\sum_{i,j=1}^n a_{ij}^2$, which gives
$$||A||_F^2=\frac{\left(b^2-1\right) a^{2 n}+a^{2 n+2}-a^2 \left(b^2 n+1\right)+b^2 (n- …
- 188k
2
votes
Accepted
Wiener-Hopf factorization of matrices
Pointers to the literature on matrix-Wiener-Hopf factorization can be found in A brief historical perspective of the Wiener-Hopf technique:
Matrix Wiener-Hopf kernels are fundamentally distinct fr …
- 188k
3
votes
Accepted
$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?
$$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
$$\big(\mathbf{M}_{\mathbf{x}}\bigr)_{nm}=- …
- 188k
35
votes
Accepted
Consequences of eigenvector-eigenvalue formula found by studying neutrinos
The OP asks about generalisations and applications of the formula in arXiv:1908.03795.
$\bullet$ Concerning generalisations: I have found an older paper, from 1993, where it seems that the same resul …
- 188k
5
votes
Accepted
Matrix equation with projection matrix
The solution for $P$ to
$$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
2
votes
Accepted
Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary en...
The $n\times n$ imaginary matrix $A$ satisfies $A^\top=-A$, so it is skew-symmetric. The Youla decomposition is
$$A=iO\Sigma O^\top,$$
where $O$ is a real orthogonal matrix and $\Sigma$ is a real bloc …
- 188k
0
votes
Generate a low-rank sparse covariance matrix
You could start from a covariance graph, and use the algorithm described in appendix A of Model-based clustering with sparse covariance matrices. This algorithm guarantees the covariance matrix positi …
- 188k
2
votes
Reference request: continuity of Cholesky factor
Numerical Analysis: A Mathematical Introduction page 295.
- 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
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