1
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2answers
276 views

How do I show this

Given invertible matrices $A,B$ and $P$ such that $A = PB$, then we say that $A$ is left equivalent to $B$. Show that left equivalence is indeed an equivalence relation.
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1answer
28 views

Equivalence involving diagonalization of block matrix

Given $T^\top L T = diag(A, B, C)$ being a block diagonalization with $L$ symmetric $A,B$ positive definite (p.d.) $T = \left( \begin{array}{ccc}I & 0 & T_2 \\ T_1 & I & T_3 \\ 0 ...
0
votes
1answer
66 views

Prove this is an equivalence relation

$A$ is related to $B$ if $M_n(A)\simeq M_m(B)$ for some integers $m$ and $n$. Clearly reflexivity and symmetry are trivial. It's transitivity that I am struggling with. Is it the case that if ...
9
votes
1answer
100 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
1
vote
4answers
358 views

Questions about the subgroups of $SU(2)$ and relevant problems?

This question is based on the invariant gauge groups in condensed matter physics( ...
1
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0answers
156 views

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked - I searched but couldn't find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) ...
1
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2answers
1k views

why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
1
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2answers
1k views

prove matrix norm equivalence

Given $A \in R^{m\times n}$, I need to prove: $$||A||_2 \le \sqrt {m}||A||_\infty$$ I have tried a number of things and I just cant seem to get it to work. Also, I need to prove: $$||A||_2 \le ...