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Results tagged with matrix-theory
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user 11260
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
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Distribution of the $\alpha$-parameter of a $2\times 2$ Haar-distributed, unitary matrix
The distribution you are seeking is:
$$P(\alpha)=\sin(2\alpha),\;\;\alpha\in[0,\pi/2]$$
The general way to derive the Haar measure for any parameterization of $U$ in terms of variables $\alpha_1,\alp …
- 188k
2
votes
Accepted
equality between the ratio trace and the determinant ratio
The equality of the arg max of the trace of the ratio and the ratio of determinants follows from the fact that each of these two maximisation problems has the same solution, given by the matrix $S$ co …
- 188k
1
vote
Eigenvector localizaiton
some pointers on eigenvector localization, mainly in the context of Perron-Frobenius theory (which in view of your earlier post seems to be what you are looking for):
Principal eigenvectors of irregu …
- 188k
2
votes
Accepted
Proving symmetry of trace function of special matrix
To check the symmetry of the trace under a given permutation of the $p_i$'s, take the corresponding permutation matrix $S$ and insert in the trace noting that $S^TS=I$ and $SWS^T=W$:
$${\rm tr}\,(AW) …
- 188k
35
votes
Accepted
Consequences of eigenvector-eigenvalue formula found by studying neutrinos
The OP asks about generalisations and applications of the formula in arXiv:1908.03795.
$\bullet$ Concerning generalisations: I have found an older paper, from 1993, where it seems that the same resul …
- 188k
2
votes
How to derive the mean of inverse-Wishart distribution?
If you start from a matrix $A$ that has a Wishart distribution with $\nu$ degrees of freedom and a $p\times p$ positive-definite scale matrix $\Sigma$ (with $\nu>p+1$), then the inverse $A^{-1}$ has t …
- 188k
2
votes
Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix $P$ in my specific (in...
I would suggest to call $E(x)=(1, x^1, x^2/2!, x^3/3!, \cdots)$ a fixed point of the linear map $E\mapsto M_\lambda\cdot E$, with $M_\lambda=\lambda P$ and scale factor $\lambda=e^{-x}$. In this way y …
- 188k
3
votes
LU decomposition for orthogonal or unitary matrices?
Here is one study of the analogue of the Cholesky decomposition for orthogonal matrices: Unconstrained representation of orthogonal matrices with application to common principle components (2019).
Re …
- 188k
15
votes
Accepted
When does the determinant distribute over addition?
let me assume $A$ is invertible, then you ask when
$$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$
so if $X$ has eigenvalues $x_i$, $i=1,2,\ldots n$, you would need
$$\prod_{i}(1+x_i)=1+\prod_i x_i$$
basically …
- 188k
1
vote
Accepted
Closed form for integral of function of a symmetric positive definite matrix
just go to a basis where $M$ is diagonal, evaluate the integral and find $Y=M^{-1}\log M$.
- 188k
6
votes
Accepted
Do matrices with only elements along the main and anti-diagonals have a name?
I am continuing in the answer box, to get this out of the "unanswered" queue. The OP asks "for a more standard terminology that is perhaps present in the literature."
The name "X-matrices" or "X-for …
- 188k
1
vote
Accepted
Bound for matrix inner product based on singular values
For a "pedagogical" proof, see A Note on von Neumann's Trace Inequality by Rolf Dieter Grigorieff.
It has been remarked in the literature that "unexpectedly, finding a
decent proof of this seemi …
- 188k
1
vote
Accepted
Eigenvalues of a block matrix with zero diagonal blocks
If you decompose $M=\begin{pmatrix} X_{q\times q}&Y_{q\times k_3}\\ (Y_{q\times k_3})^{\rm T}&0_{k_3\times k_3}\end{pmatrix}$ into four block matrices, with $q=k_1+k_2$, then the determinant equals
$$ …
- 188k
3
votes
Accepted
Singular value decomposition of truncated discrete Fourier transform matrix
Let me insert a factor $N^{-1/2}$, so that the Fourier transform is unitary:
$$U_{mn}=N^{-1/2}e^{-2\pi i(m-1)(n-1)/N},\quad m,n=1,\ldots,N.$$
We truncate the $N\times N$ matrix $U$ to the $k\times k$ …
- 188k
3
votes
Local differentiability of eigenvalues and eigenvectors of a real symmetric matrix
Theorem (1.1) of Perturbation theory for normal operators is likely what you are looking for.
See also Differentiable perturbation of unbounded operators.
If you would order the eigenvalues by their m …
- 188k