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2d
answered Find the elementary divisors of a matrix given its characteristic and minimal polynomials
2d
comment Find the elementary divisors of a matrix given its characteristic and minimal polynomials
Just a point about the formulation: elementary divisors are associated to a module over a PID, not to a matrix. No doubt what is meant here are the elementary divisors of $K^n$, viewed as $K[X]$-module with the indeterminate $X$ acting as multiplication by the matrix$~A$ (with $K=\Bbb Q$, and $n=16$ if I've computed $\deg(p_A)$ correctly).
2d
revised Minimal polynomial for diagonalizable operators
edited tags
2d
revised Kernel decomposition as direct sum, related to minimal polynomial of a linear operator
added 471 characters in body
Jul
24
comment I have to show $A$ commute with a non-central matrix whose nullity is at least $\dfrac {(p+1)}{2}$.
What about taking the zero matrix? It commutes with everyone and its nullity could not be larger.
Jul
24
comment Determine the null space of a linear map
The set of polynomials of degree $k$ is not a vector space. Did you mean polynomials of degree at most $k$?
Jul
23
awarded  Revival
Jul
23
revised Rudin's equivalent in Linear Algebra
minor fiddling
Jul
22
comment Why do we show that structures aren't isomorphic by exhibiting a property not shared by one of them?
Note that in your first paragraph you need to require that $f$ is a bijection; if not all bets are off.
Jul
22
comment If I buy 2 lottery tickets do I double my chance of winning?
It seems quite evident in the question that the winning combination is drawn independently of any tickets sold (and so quite possibly nobody gets the jackpot). So the "another lottery" stuff is just a needless diversion; don't add confusion that is absent in the question.
Jul
22
comment Matrix,Linear algebra,polynomial,finite field,notation
Because for $x\in[n]$ those matrix entries were already defined; the purpose here (though very badly expressed) is to define those entries for other values $x$ by polynomial interpolation.
Jul
22
answered Matrix,Linear algebra,polynomial,finite field,notation
Jul
22
comment Matrix,Linear algebra,polynomial,finite field,notation
The phrase "defined in the unique wys such that" is plain nonsense. I can define $(D_A(x))_{j,k}$ to be always $0$ if I feel so inclined. Or I could define it to be $(X+j)(X+k)$. Or whatever. These are both (all) polynomials of degree at most $n$ (assuming $n\geq2$) so there is no uniqueness.
Jul
22
comment Matrix,Linear algebra,polynomial,finite field,notation
There is a problem wth the question in that $D_A(i)$ is defined (only) when $i\in[n]$, but then $D_A(x)$ is used where $x\in[n]$ is explicitly excluded! It is not so clear what $F$ is where $x$ lives; maybe the finite field (whose elements I would have a hard time matching up with column indices $i$), but in any case if some $x\in F$ would happen to also be a column index $i\in[n]$, then that value is forbidden.
Jul
22
revised Matrix,Linear algebra,polynomial,finite field,notation
you probably meant quotation style, not verbatim
Jul
22
revised Rudin's equivalent in Linear Algebra
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Jul
22
answered Rudin's equivalent in Linear Algebra
Jul
18
comment Do we have $\mathbb{C}[SL_n] = \oplus_{\lambda, \text{ht}(\lambda)\leq n} V_{\lambda} $?
It would be suprising if one had equality both times (the LHSs look different). Please share your own thoughts about these questions with us. Without knowing where the questions came from and what you know already, it will be inpossible to answer the questions at the appropriate level.
Jul
14
comment Does there exist such an invertible matrix?
Your question appears to use only the ring $A$ of polynomials in $x$, and not mention rational functions at all. Did you forget to do anything with them?
Jul
14
awarded  Necromancer