Wikipedia:Reference desk/Archives/Mathematics/2019 November 15

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November 15

linear algebra: the adjoint operator of the inner product of integral

Given f and g polynomial functions R->R with the height degree 1, and T(bx+a)=ax+b and the inner product integral(f,g).

find T*.

I thought that I should just take the transpose matrix of T but it seems an incorrect answer. Why? I thought that on R T* must be the transpose — Preceding unsigned comment added by 77.127.109.73 (talk) 20:17, 15 November 2019 (UTC)

A hint (although it's been a while and I'm a bit rusty with this stuff, so take it with a grain of salt): Using the transpose will only work if you've written the matrix of your linear map with respect to an orthonormal basis. So if you wrote the matrix of T with respect to, say, the basis $\{1,x\},$ that won't necessarily work. Try to apply Gram-Schmidt to get an orthonormal basis first. –Deacon Vorbis (carbon • videos) 20:54, 15 November 2019 (UTC)

Alternatively (if you don't know that stuff), you can still work with the above basis directly, which is easy enough for a 2-dimensional vector space, although will get worse in higher dimensions. You haven't told us what interval the inner product is with respect to, so I'm just going to assume $[0,1]$ here; you can adjust if needed. You want to find out what $T^{*}$ does to each basis vector, and you can find that out by testing against the inner product with each basis vector. So

$\langle T^{*}1,1\rangle =\langle 1,T1\rangle =\langle 1,x\rangle =\int _{0}^{1}x\,dx=1/2.$

Likewise, $\langle T^{*}1,x\rangle =1.$ If you write $T^{*}1$ as, say, $ax+b,$ then you find that $\langle T^{*}1,1\rangle =\int _{0}^{1}(ax+b)\,dx=a/2+b.$ Do the same for $\langle T^{*}1,x\rangle$ and you'll have a system of two equations in two unknowns. Solve it, and you know what $T^{*}1$ is. Then just repeat for $T^{*}x$ and you'll have all you need. –Deacon Vorbis (carbon • videos) 21:15, 15 November 2019 (UTC)

Say the inner product integral is $(f,g)_{\mu }:=\int _{\mathbb {R} }fgd\mu$, and put :

$m_{k}:=\int _{\mathbb {R} }x^{k}d\mu$

for integer $k=0,..,2$. For $f=u_{0}+u_{1}x$ and $g=v_{0}+v_{1}x$ we have $(f,g)_{\mu }=m_{0}u_{0}v_{0}+m_{1}(u_{0}v_{1}+u_{1}v_{0})+m_{2}u_{1}v_{1}$. That is, in the basis $x^{0},x^{1}$ the inner product on these linear polynomials is represented by the matrix

$M:={\begin{bmatrix}m_{0}&m_{1}\\m_{1}&m_{2}\end{bmatrix}}.$

The operator $T$ has the matrix

$T:={\begin{bmatrix}0&1\\1&0\end{bmatrix}}.$

We have, wrto the scalar product or $\mathbb {R} ^{2}$

$(f,Tg)_{\mu }=Mu\cdot Tv=TMu\cdot v=M(M^{-1}TM)u\cdot v$

whence the transpose of T (wrto the inner product integral) is represented in the scalar product of $\mathbb {R} ^{2}$ by

$M^{-1}TM={\begin{bmatrix}m_{0}&m_{1}\\m_{1}&m_{2}\end{bmatrix}}^{-1}{\begin{bmatrix}0&1\\1&0\end{bmatrix}}{\begin{bmatrix}m_{0}&m_{1}\\m_{1}&m_{2}\end{bmatrix}}={\frac {1}{m_{0}m_{2}-m_{1}^{2}}}{\begin{bmatrix}m_{1}(m_{2}-m_{0})&m_{2}^{2}-m_{1}^{2}\\m_{0}^{2}-m_{1}^{2}&m_{1}(m_{0}-m_{2})\end{bmatrix}}$

So the final expression for the transpose is

$T^{*}(a+bx)={\frac {m_{1}(m_{2}-m_{0})a+(m_{2}^{2}-m_{1}^{2})b}{m_{0}m_{2}-m_{1}^{2}}}\;+\;{\frac {(m_{0}^{2}-m_{1}^{2})a+m_{1}(m_{0}-m_{2})b}{m_{0}m_{2}-m_{1}^{2}}}\,x$

— Preceding unsigned comment added by PMajer (talk • contribs) 11:17, 17 November 2019 (UTC)

There are a few problems here. You keep referring to the "transpose", but you seem to mean the adjoint. The whole confusion in the first place was why you couldn't simply use the transpose of T (because the basis wasn't orthonormal). You're also using measure notation in your integrals, while the question is elementary enough that it's very possible (even likely) that the OP has no idea what that is. You also sneakily make the claim that $Mu\cdot Tv=TMu\cdot v.$ This is true, but only because T is symmetric (wrt the standard basis); any other operator and you would have had to use the transpose of T instead. –Deacon Vorbis (carbon • videos) 15:01, 17 November 2019 (UTC)

Yes there are a few problem here. pma 22:40, 17 November 2019 (UTC)

...mainly comprehension problems though. Some comments may help: A) The OP did not specify what is their inner integral product, nor an interval: I wished to point out that the choice of a measure is not relevant for the answer. On this issue: B) I used to answer questions here since 2008 for several years on a daily basis; from my experience it is better to wait for a request of further details from the OP or anybody else, rather than giving a 100% complete treatise including all related subjects C) Yes, the question is elementary, but I'd like to show the right way to solve these issues, if the OP or anybody else is interested (and who is not interested is welcome as well, of course) C) Yes, I mean the (real, Hilbert) transpose. You do not need an orthonormal basis to define a transpose, it is defined by means of a scalar product. D) Unfortunately, language and notations are not uniform in this topic, possibly because it is used in all fields of mathematics; in my experience of professional mathematician I can tell you that "adjoint" and "transpose" are very often used as synonymous; this is also confirmed by the wiki article btw. pma 13:09, 18 November 2019 (UTC)