For questions about adjoints, in the category-theoretic or inner-product-space sense, as well as about adjugate matrices.

learn more… | top users | synonyms

1
vote
0answers
34 views

$\operatorname{Im} A = (\operatorname{ker} A^*)^\perp$

Let $A:\mathbb{R}^m \to \mathbb{R}^n$ be a linear transformation. We know that there is a unique transformation $A^*:\mathbb{R}^n \to \mathbb{R}^m$ such that $$\langle Ax,y\rangle = \langle x,A^*y ...
3
votes
0answers
33 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
2
votes
0answers
35 views

Linear map is diagonalizable iff its adjoint is diagonalizable

Problem Let $V$ be a finite inner product space and let $T:V \to V$ be a linear transformation. Prove that $T$ is diagonalizable if and only if the adjoint transformation $T^{*}$ is diagonalizable. ...
1
vote
0answers
25 views

The existence of adjoint operation on Banach space

I have a question about adjoint operator. I have known that bounded linear operator on Hilbert space has a unique adjoint operator, but I am wondering whether there is similar existence result about ...
1
vote
2answers
45 views

How to find T* if the product inner space is not explicitly

I have a little question. I was studying for my Linear Algebra test and I tryed the following problem: If $T\in\mathcal{L}(K^n)$ is such that $T(z_1,\ldots,z_n)=(0,z_1,\ldots,z_{n-1})$. Determine ...
0
votes
0answers
18 views

Regarding Adjugate of a Module Homomorphism

$\newcommand{\adj}{\text{adj}}$ I am trying to understand the concept of adjugate of an endomorphism of free module of finite rank as discussed here in Section 8. Let $M$ be a free module over a ring ...
5
votes
2answers
83 views

Left adjoint of the forgetful functor $\mathsf{Grpd} \to \mathsf{Cat}$?

I've heard that there is a left adjoint to the forgetful functor $\mathsf{Grp} \to \mathsf{Mon}$. I wonder if there is also a left adjoint $F : \mathsf{Cat} \to \mathsf{Grpd}$ to the forgetful functor ...
0
votes
0answers
28 views

Singular Value Decomposition problem

Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$ and let $V$ be any finite-dimensional inner product space over $\mathbb{F}$. Let $T: V \to V$ be Linear transformation. Prove that $T$ and $T^*$ have ...
2
votes
2answers
27 views

Adjoint of $\lambda I - T$

Given a selfadjoint (maybe unbounded) operator $T$ on a Hilbert space $H$, I want to calculate the adjoint of $\lambda I - T$ for a $\lambda \in \Bbb C$. I am tempted to argue as follows: ...
0
votes
0answers
17 views

Square root of Differential Operator Self-Adjoint

If $H=L^2(0,2)$ $Aw=w^{(4)}, D(A)=H^4(0,2)\cap H^2_0(0,2)$ and $Bv=-(d(x)v')', D(B)=\{v\in H^1_0(0,2)\mid dv' \in H^1(0,2)\}$ where $$d(x)=\begin{cases}1\text{ if }0<x<1 \\ 2\text{ if ...
3
votes
1answer
90 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
0
votes
1answer
24 views

In what condition we have $(K^{-1})^\ast = (K^\ast)^{-1}$?

Suppose $X$ $Y$ are two finite dimensional Hilbert space. Assume $K$: $X\to Y$ is linear. My question is, in what condition of $K$ that $$(K^{-1})^\ast = (K^\ast)^{-1}?$$
1
vote
1answer
24 views

Two “adjunct” (quasi-inverse) functions

Let $A$, $B$ be fixed sets. What "means" the formula $Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$ for functions $\alpha:\mathscr{P}A\rightarrow\mathscr{P}B$ and ...
3
votes
1answer
60 views

Adjoint squares

I'm reading Mac Lane's Categories for the Working Mathematician and I'm having some trouble with exercise 5 in part IV.7. To avoid introducing adjoint squares I will only formulate the question in ...
1
vote
1answer
40 views

Coadjoint action $\operatorname{Ad}^*_\phi(h)$ respects coproduct $\Delta$?

In Majid's quantum group primer at the beginning of Chapter 3, page 18, he's proving that if $H'$ and $H$ are dually paired bialgebras or Hopf algebras, the coadjoint action $$ ...
-1
votes
1answer
37 views

how to find inverse of a matrix

How to find the inverse of a 4x4 order matrix using adjoints for example $$A=\begin{pmatrix} 2 & -6 & -2 & -3 \\ 5 &-13 &-4 &-7 \\ -1 & 4& 1& 2 \\ 0 & 1 ...
5
votes
2answers
137 views

A proof that right adjoints preserve limits?

Assume that categories $\mathscr{B}$ and $\mathscr{C}$ have all limits of shape $\mathbf{J}$. Then there's a slick proof that if $G\colon \mathscr{C} \to \mathscr{B}$ is a right adjoint, $G \circ ...
1
vote
2answers
46 views

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible. I cannot think of an example to save my life. Any solutions/hints are greatly ...
3
votes
1answer
76 views

How to prove this is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to ...
0
votes
1answer
39 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
4
votes
3answers
82 views

Is this operator $A = \pmatrix{1&1\\0&1}$ self-adjoint?

Is this operator $$A = \pmatrix{1&1\\0&1}$$ self-adjoint? I think not, because $$\pmatrix{1&1\\0&1}^T\neq A$$ where $T$ is the transposition of the matrix. What do you all think?
0
votes
1answer
12 views

Proving the adjoint nature of operators using Hermiticity

How can the fact that $\hat x$ and $\hat p$ are Hermitian be used to prove that $\hat x - \frac{i}{m \omega} \hat p$ and $\hat x + \frac{i}{m \omega} \hat p$ are adjoints of each other?
3
votes
2answers
38 views

When is the restriction of a normal operator not normal?

I was proving the spectral theorem for normal operators on finite-dimensional complex vector spaces today during a test, when I arrived at the point in which If $T\in\operatorname{End}(V)$ is ...
0
votes
1answer
26 views

Work out the adjoint of $T(x,y) = (y,-x)$

this seems like a simple question but I don't understand it. We define a transformation $T(x,y) = (y,-x)$. We want to work out what the adjoint is. I know the answer: $T^*(x,y) = (-y,x)$ but how? ...
2
votes
1answer
58 views

Confirm my understanding of adjoints

adjoints seem REALLY important and useful so I don't want to move onto the next topic without really understanding them; I have too many a times moved on and been lost because I don't have the ...
1
vote
3answers
26 views

Find the adjoint of this non-standard inner product space

I'm really blanking out (a lot of late nights these past 10 weeks). The point of the exercise I'm about to type up is to show that the adjoint structure may possibly change when the inner product ...
0
votes
1answer
26 views

Adjoint of $T_A = Ax$

Is it true that if $T_A(x) = Ax$ then $T^*_A(x) = A^*x$? I tried to prove this for the standard inner product $$ \newcommand{\innp}[2]{\left\langle #1,#2 \right\rangle} \innp{Ax}{x} = ...
0
votes
1answer
35 views

Proving facts about adjoints

Let $F$ denote $\mathbb R$ or $\mathbb C$. Let $T : V → W$, $S : V → W$ and let $R: U → V$ be linear transformations between inner product spaces $U$, $V$, $W$ over $F$. Verify the following facts: ...
0
votes
2answers
39 views

Prob. 8, Sec. 3.10 in Kreyszig's functional analysis book: An isometric linear operator has its adjoint as its left inverse

Let $H$ be a Hilbert space, and let $T \colon H \to H$ satisfy $$\langle Tx, Tx \rangle = \langle x, x \rangle \ \mbox{ for all } \ x \in H.$$ Then $T$ is bounded and norm $\Vert T \Vert = 1$ (unless ...
1
vote
1answer
22 views

Prob. 6, Sec. 3.10 in Kreyszig's functional analysis book: Powers of self-adjoint operators

Let $H$ be a Hilbert space. If $T \colon H \to H$ is a bounded self-adjoint linear operator and $T \neq 0$, then $T^n \neq 0$ for all $n \in \mathbb{N}$. How to show this? I've managed to show ...
1
vote
1answer
62 views

Prob. 10, Sec. 3.9 in Kreyszig's functional analysis book: The null space and adjoint of the right-shift operator

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$, let $T \colon H \to H$ be defined as follows: Since span of $(e_n)$ is dense in $H$, for every $x \in H$, we have $$x = ...
1
vote
0answers
28 views

Prob. 2, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Inversion and adjointness

Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bijective bounded linear operator whose inverse is bounded. Then how to show that $(T^*)^{-1}$ exists and $$(T^*)^{-1} = (T^{-1})^*?$$ My ...
4
votes
0answers
30 views

Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = ...
7
votes
1answer
93 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
2
votes
2answers
79 views

Right adjoint of covariant hom functor

I've constructed the left adjoint of the functor $\mathbf{Hom(A, -)}: \mathbf{Sets} \to \mathbf{Sets}$. Now I'm trying to prove that the functor does not have a right adjoint, but I'm not sure how to ...
0
votes
1answer
29 views

What is the adjoint of an inverse matrix? [duplicate]

What is the adjoint of an inverse matrix? Is $(T^{-1})^{*} = (T^{*})^{-1}$?
3
votes
2answers
45 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
0
votes
2answers
49 views

proof of all integer entries in an inverse matrix

"Prove that if $det(A)=1$ and all the entries in $A$ are integers, then all the entries in $A^{-1}$ are integers." I began by setting up the adjoint method for finding the inverse. $A^{-1} = \cfrac ...
12
votes
2answers
166 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
3
votes
1answer
65 views

$T \in B(X,Y)$ is an isometry if and only if $T^*$ is an isometry

I would like to prove that $T \in \mathscr{B}(X,Y)$ is an isometry of $X$ onto $Y$ if and only if $T^*$ is an isometry of $Y^*$ onto $X^*$. I am not really sure what to do. I started the argument as ...
0
votes
2answers
49 views

Product of compact, bounded and self adjoint operator.

$T \in B(H)$, and $T = S^2$ for some self adjoint operator $S \in B(H)$. I need to prove that T is compact if and only if S is compact. If S is compact, it is easy to show that T is compact since S ...
2
votes
1answer
79 views

Adjoint operator on Banach space

Suppose $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded linear operator. Show that $T$ is an isometric isomorphism if and only if its adjoint $T^*$ is also an isometric isomorphism. Given an ...
5
votes
1answer
128 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
0
votes
1answer
24 views

About the self-adjoint extension of an operator.

Let $B$ be a selfadjoint extension of an operator $A$ on a Hilbert space $H$. Let $\varphi \in \ker(A^\ast-z_0)$. Then i want to show that $\varphi + (z- z_0)(B-z)^{-1} \varphi \in \ker(A^\ast-z)$. I ...
0
votes
0answers
45 views

Adjoint of a 3x4 matrix

How do I find the adjoint of this matrix? I am familiar with finding the adjoint of an $n x n$ matrix, but this has thrown me. $$A= \left( \begin{array}{ccc} 1&-1&0\\ 0&0&1\\ ...
3
votes
1answer
76 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
0
votes
1answer
13 views

Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
0
votes
0answers
118 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
4
votes
2answers
92 views

Adjoint functor to an R-algebra only “remembering” itself as a ring

I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let $$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$ be the functor that sends ...
1
vote
1answer
61 views

let $\dim(ker (A - \lambda I)) = 1$. why is $adj(A - \lambda I) \ne 0$

Let $\lambda$ is eigenvalue of $A$ and $\dim(\ker (A - \lambda I)) = 1$.($\lambda$ has geometric multiplcity one) why is $\text{adj}(A - \lambda I) \ne 0$?