Tagged Questions
12
votes
2answers
473 views
The infinite-dimensional limit of sequence of solutions of linear equations when the number of equations goes to infinity
Suppose we have an infinite-dimensional real vector $y=(y_1,...)$. Suppose we have an infinite-dimensional real matrix $C=(c_{ij})$, $i,j\in\mathbb{N}$. Let $C^k$ be a submatrix of $C$, ...
8
votes
1answer
71 views
*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$
I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form
$A\in ...
6
votes
2answers
937 views
How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$?
Statement: If $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$ and vice versa.
One way of the proof.
We have $B(B^{-1}A ) B^{-1} = AB^{-1}. $ Assuming $ ...
6
votes
1answer
99 views
Cauchy Integral Formula for Matrices
How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
6
votes
0answers
137 views
Proof that the set of doubly-stochastic matrices forms a convex polytope?
Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
5
votes
1answer
60 views
Functions space of discrete space: how does taking quotients lead to noncommutativity?
It is pointed out in Geometry from the spectral point of view the following:
If one considers a discrete space, say, the two-point space $\{1,2\}$,
after identifying its points $X=\{1,2\}/\sim$, the ...
5
votes
1answer
99 views
Second conjugate operators and their representations
We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,n\geq 1}$ which acts on a sequence $a=[a_1, a_2, a_3, \ldots ]\in c_0$ in the same way as usual ...
4
votes
2answers
69 views
Is Householder orthogonalization/QR practicable for non-Euclidean inner products?
The question
Is there a variant of the Householder QR algorithm to orthonormalize a set of vectors with respect to an inner product if no orthonormal basis is known a priori?
Background
Let's ...
4
votes
1answer
421 views
Cauchy interlacing theorem - how to complete this proof
I am looking for a proof using the min-max principle.
Wikipedia seem to provide just that: http://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem
But this part seems to be wrong:
...
4
votes
1answer
286 views
Matrix Analysis in $\mathbb{C}$
1.Let $A \in GL_n(\mathbb{C})$. Show that $\det(I+A)=1+\operatorname{tr}(A)+ \epsilon(A)$ where Modulas of epsilon(A) by norm of A=0 as A tends to 0,for any matrix norm.
If I define J(A)= det(A) for A ...
3
votes
1answer
50 views
Norm of a linear transformation
Let $T:\mathbb R^2\to \mathbb R^2$ be given by the matrix $\begin{pmatrix}a&b\\ c& d\end{pmatrix}$. Let $u:=a^2+b^2+c^2+d^2+2(ad-bc)$ and $v:=a^2+b^2+c^2+d^2-2(ad-bc)$.
I need to show that ...
3
votes
3answers
189 views
Proving two results about the spectral radius
How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces?
Theorem 1.
Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon ...
3
votes
1answer
86 views
Let $ \rho(P)$ be the spectral radius of $P$. Show $ \rho( \dfrac{P}{ \rho(P) + \epsilon } ) < 1 \text{ for all } \epsilon >0. $
Let $P$ be a square matrix and $ \rho(P)$ the spectral radius of $P$. How to show
\begin{align} \rho\left( \dfrac{P}{ \rho(P) + \epsilon } \right) < 1 \text{ for all } \epsilon >0. \end{align}
...
3
votes
1answer
94 views
Skew symmetric matrix decomposes
I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $ R $ such that $ R^T A R = M$, ...
3
votes
0answers
32 views
Supremum over unitary group action
Let $A$ and $B$ are two given Hermitian operators on matrix algebra $M_n(\mathbb{C})$. $A$ is positive semi-definite with unit trace. I want to know the general method for calculating the following ...
