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Results tagged with matrix-analysis
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user 8799
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.
17
votes
Accepted
Counting eigenvalues without diagonalizing a matrix
Here is an efficient method.
First of all, I must quote that diagonalizing $M$ is not a method, because there is no explicit way to carry this out. It amounts to calculating the roots of a polynomial …
- 52.3k
17
votes
Accepted
The abc-conjecture as an inequality for inner-products?
The matrix $L_n$ is positive definite.
Proof. The matrix $G_n$ with entries ${\rm gcd}(a,b)$ is positive definite because of $G=D^T\Phi D$ where $\Phi={\rm diag}(\phi(1),\ldots,\phi(n))$ ($\phi$ the …
- 52.3k
15
votes
Accepted
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ ma...
Elementary proof. The linear space $E$ spanned by $SO_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO_n$. Let $M=SR$ be a polar decompositi …
- 52.3k
9
votes
Triangularizing a matrix with function entries
You cannot triangularize smoothly the parametrized matrix
$$A(z)=\begin{pmatrix} 0 & 1 \\\\ z & 0 \end{pmatrix}$$
about $z=0$. The eigenvalues, square roots of $z$ aren't smooth and, above all, cannot …
- 52.3k
9
votes
Accepted
One question on block-circulant matrices
The formula for the specific case is
$$\det K=\det(A+B+C+D)\det(A-B+C-D)\det(A+iB-C-iD)\det(A-iB-C+iD).$$
More generally, for a block-circulant matrix with $n$ square blocks $A_0,\ldots,A_{n-1}$, the …
- 52.3k
8
votes
Accepted
A matrix inequality involving the Hilbert-Schmidt norm
Suppose $Q$ is such a form. Write that the mean value of $Q$ over the unit sphere is non-negative. You obtain
$$-\frac12\sum_ia_{ii}^2+\frac12\sum_{i < j}a_{ii}a_{jj}-9\sum_{i < j}a_{ij}^2\ge0.$$
This …
- 52.3k
7
votes
Accepted
Argument principle for matrices
You can write instead
$$\frac1{2i\pi}\int_Cg(z){\rm tr}(f'(z)f(z)^{-1})dz.$$
Now use the formula
$${\rm tr}(f'(z)f(z)^{-1})=\frac1{\det f(z)}\,(\det f(z))'.$$
And conclude with the formula of residues …
- 52.3k
7
votes
Accepted
When are two binary matrices simultaneously equivalent to their transpose?
Clearly, a necessary condition is that for every word $w$ in two letters, one has
$${\rm Tr}\,w(A^t,B^t)={\rm Tr}\,w(A,B).$$
Equivalently,
$${\rm Tr}\,\hat w(A,B)={\rm Tr}\,w(A,B),$$
where $\hat w$ is …
- 52.3k
5
votes
A question of invertibility of matrices
What about
$$A=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\qquad B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\quad ?$$
- 52.3k
4
votes
Fast Upper Triangular Matrix Exponentiation
When I teach the exponential of matrices, I tell the students that the converging series is not a practical tool for calculation. It is way better to solve the differential equation. This turns out to …
- 52.3k
4
votes
Accepted
Relation between Frobenius norm, infinity norm and sum of maxima
The answer is Yes. This is not really a problem about matrices. The best way to analyse it is to rewrite it in terms of the row vectors $u_i\in{\mathbb C}^n$. Let me denote $\|\cdot\|_p$ the $\ell^p$- …
- 52.3k
4
votes
Accepted
Question on density of certain set of matrices
It suffices to check whether $B^{-1}S=:S'$ has measure zero in $B^{-1}Q=:Q'$. We have
$$Q'={\bf Sym}_n(\mathbb R),\qquad S'=\{{\bf Sym}_n(\mathbb R)|B(\Sigma+\Sigma^3)\in{\bf Sym}_n(\mathbb R)\}.$$
Th …
- 52.3k
3
votes
Logarithms of matrices in the disk-algebra
It seems that $\Delta(z)$ is the exponential of an holomorphic $M(z)$. Using the eigenvalues and eigenvectors of $\Delta$, I find
$$M(z)=\frac\mu{\sqrt{z(z+4)}}\begin{pmatrix} -z & 2 \\ 2z & z \end{pm …
- 52.3k
3
votes
system of homogeneous matrix equations
A partial answer: Let $\Sigma$ denote the manifold $x^n+y^n=0$. Away from $\Sigma$, the equation and the fact that the roots of the polynomial $X^n-x^n-y^n$ are simple, tell you that $xA+yB$ is diagon …
- 52.3k
3
votes
Accepted
A sum of eigenvalues
The answer is yes, because your function (let me call it $f$) is the maximum of convex functions. As such, it is convex. The formula :
$$f(X)=\max\left(0,\max_{1\le r\le n}\sum_{j=1}^r\lambda_j(X)\rig …
- 52.3k