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

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1answer
22 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
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1answer
67 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
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1answer
19 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 ...
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0answers
27 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
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1answer
58 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
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1answer
11 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, ...
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0answers
26 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
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2answers
65 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 ...
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1answer
51 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 $\text{adj}(A - \lambda I) \ne 0$?
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1answer
35 views

A and B are similar then adj(A) and adj(B) are similar [closed]

If $A,B \in {M_n}$ are similar, why $adj(A)$ and $adj(B)$ are similar?
0
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1answer
30 views

Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
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2answers
31 views

relation eigenvalue and adj(A-λI)

Let $A$ be a matrix in $\mathbb C^{n×n}$, let $λ$ be an eigenvalue of $A$ with eigenvector $x$. Why is there some $y \in \mathbb C^n$ such that $adj(A−λI)=x{y^*}$?
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1answer
45 views

Showing that if $A$ is closed, then $A^\ast A$ is self-adjoint

Let $A$ be a closed linear operator on a Hilbert space $H$. Then I want to show that $B = A^\ast A$ is self-adjoint. Now, $B$ is positive, i.e. $\langle f, B f \rangle \geq 0 \forall f \in D(B)$. ...
0
votes
1answer
26 views

Find a general 2x2 matrix where A = adj(A)

I know how to find the adjoint of 2x2 matrix but I'm at a loss for finding a general 2x2 matrix where A = adj(A). Thanks for your help!
0
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4answers
59 views

Calculate inverse of matrix

If $$A=\begin{bmatrix} -5 & 1 & 0 & 0\\ -19 & 4 & 0 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 3 & 5\\ \end{bmatrix}, $$ how do I calculate $A^{-1}$? Is there any ...
0
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1answer
20 views

Adjoint operators in Hilbert space

Consider the linear and bounded operators $X$ and $Y$on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot,\cdot \rangle$. How can I show that $$ \langle XY \boldsymbol{v}, ...
2
votes
1answer
85 views

Integration by parts to find the adjoint operator

On the interval $(0,1)$ consider the differential operator $Lu=u''''+u'$ with boundary conditions $u(0)+u'(1)=u(1)+u'(0)=0$ $2u(0)+u''(1)=2u(1)+u''(0)=0$ $(1)$ I want to find the adjoint ...
0
votes
0answers
14 views

prove operator is normal

Consider, $$A = \begin{bmatrix} 0 & I \\ c^2\frac{\partial^2}{\partial x^2} &-c_d I \end{bmatrix}$$ I want to construct an inner product space where AA* = A*A. That is A is a normal ...
2
votes
2answers
35 views

Bounded Linear Operator and the Adjoint

Let $S$ be a linear operator with dense domain $\mathcal{D}(S)$ in the Hilbert space $\mathcal{H}$. Assume that the domain $\mathcal{D}(S)$ belongs to a larger domain, namely $\mathcal{D}(S) \subset ...
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0answers
55 views

Functors adjoint from both sides

If F is left adjoint to G and G is left adjoint to F, does it imply anything nice about these functors?
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1answer
37 views

Question about the notation $S \subset T$ ,where $S$ and $T$ are operators

I want to prove that if $S\subset T$. Then $T^{*}\subset S^{*}$. But what does $S\subset T$ mean? $S$ and $T$ are operators and not sets.. :/
2
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3answers
73 views

Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
3
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1answer
35 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
0
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1answer
27 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
2
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1answer
30 views

adjoint representations

I am trying to work out the adjoint representations of $$H=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), X = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} ...
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0answers
132 views

Adjoint of an adjoint linear map

My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector ...
2
votes
1answer
57 views

A functor that has both left and right adjoints

What can we say about a functor that has both left and right adjoints? I vaguely recall hearing that it is then an equivalence of category. Is it true? If not, then under what conditions it is true? ...
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0answers
23 views

Proof of inverse matrix element explicit formula [duplicate]

There is a matrix: A, and exists an inverse matrix: A^-1 which elements are b. (b)ij = adj(Aji) / det(A) What is the proof of this equation?
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1answer
53 views

Existence of adjoint of the inverse

Let $H$ be a Hilbert space over $\mathbb{F}$ and $V$ be an inner product space over $\mathbb{F}$. Let $T:H\rightarrow V$ be a bounded linear bijection. If $V$ is a Hilbert space, then the open ...
3
votes
2answers
145 views

Question on adjoint functors

Can someone provide me an enlightenment on the following three statements? (I stumbled on them at the part dealing injective modules in a text of homological algebra.) 1) Let $F \dashv G \colon ...
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2answers
132 views

Tensor-Hom Adjunction In Monoidal Categories?

Is there a generalization of the tensor-hom adjunction to monoidal categories, or is it a special property of $\mathsf{Mod}$-$R$?
2
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1answer
52 views

Adjunctions via Reflections and the Axiom of Choice

I have met two ways of defining adjunctions: via the triangle identities, and via reflections. Proposition 3.1.2 Let $F:\mathsf A \rightarrow \mathsf B$ be a functor and $B$ an object of $\mathsf ...
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2answers
74 views

Prove that $T^*$ is injective iff $ImT$ Is dense

Let X,Y be two normed spaces, and $T:X\rightarrow Y$ a bounded linear operator. prove that the adjoint operator $T^*$ ($T^*f(x)=f(Tx)$ is injective iff $ImT$ is dense any help would be great guys. I ...
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0answers
29 views

Change of base - Hermitic matrices

This exercise comes from a university exam (http://www.ubacs.com.ar/foro/viewtopic.php?f=67&t=3079, link in spanish). I'll copy it in english for everyone. It's #3: We define in $C^{n×n}$ the ...
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0answers
38 views

Adjoint linear operators and inner products question; why does $\langle T(x),T(x)\rangle =\langle T^*T(x),x\rangle $?

I have seen this multiple times in my textbook; $\langle T(x),T(x)\rangle=\langle T^*T(x),x\rangle$; why is this true? I know the definition of adjoint is if $\langle x,T(y)\rangle=\langle ...
1
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1answer
97 views

Kernel of adjoint and orthogonal complement images

Alright, suppose we are given $V$, a finite dimensional inner product space, and a linear map, $T:V \rightarrow V$, with its corresponding adjoint, $T^\star :V \rightarrow V$. I want to show: ...
3
votes
2answers
77 views

Sequences or 'chains' of adjoint functors [duplicate]

Suppose we have (some categories and some functors such that) $F_1$ is left adjoint to $G_1$, $G_1$ left adjoint to $F_2$, $F_2$ left adjoint to $G_2$. Will $F_1$ then be equal to $F_2$ (and $G_1$ to ...
1
vote
1answer
38 views

Green's operator, differential forms

In "Foundations of Differential Manifolds and Lie Groups" by Frank Warner on page 225 there is defined Green's operator: $G: E^p(M) \rightarrow (H^p)^{\perp}$ by setting $G(\alpha)$ to equal the ...
3
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2answers
182 views

More than one pair of “nice” adjoint functors between different concrete categories

Though adjoint functors provide a universal description for many concrete mathematical constructions, these constructions usually revolve around finding a single "canonical" way to transform one type ...
3
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1answer
70 views

Schur decomposition of a matrix with distinct eigenvalues is almost unique

Let $M\in \mathbb C^{n,n}$ have $n$ distinct eigenvalues, and let $U_1, U_2$ be two Schur-forms of $M$. Show that if $U_1, U_2$ have equal diagonals, there is a hermitian diagonal matrix $Q$ such ...
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0answers
17 views

Adjoint of Polynomial?

I understand adjoint of matrices. Say I have a transformation to the standard basis [1,x,$x^2$]. Given: T(p(x)) = p(x+i) Find T*(p(x)) for all. How do I apply the adjoint transformation to the ...
0
votes
1answer
32 views

Is it always the case that a free construction satisfies this universal property?

this might be a stupid question, but I'm not sure if this is true (at least in some class of cases). Let $F : \mathcal{C} \rightarrow \mathcal{D}$ be left adjoint to an inclusion $\mathcal{D} ...
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2answers
48 views

Finding the determinant of a matrix given the adjoint

My attempt: Knowing that $$A(AdjA) = IdetA$$ I took the determinant on both sides: $$det(A)det(AdjA) = det(det(A))$$ So, $$det(A)det(AdjA) = (det(A))^3$$ $$det(AdjA) = (det(A))^2$$ $$det(A) = ...
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2answers
92 views

Proving that $\chi_{T^*}=\overline{\chi_T}$ and $m_{T^*}=\overline{m_T}$ (characteristic and minimal polynomials of adjoint map)?

For a linear map $T:V\to V$ where $V$ is a finite dimensional inner product space over $\mathbb{C}$, I know the result $\chi_{T^*}=\overline{\chi_T}$ (where $T^*$ is the adjoint map for $T$). My ...
0
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1answer
55 views

Adjoint of a matrix and inverse of a matrix

As everyone know that we can use a matrix $A$ to represent an operator $T$. The adjoint of a matrix $A$ is denoted as $A^*$, which takes complex conjugate of $A$ and then transpose. My problem ...
0
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1answer
21 views

Show $\sigma(T)=\sigma{(\overline{T^{*}})}$

Let $T \in B(H)$ be a bounded operator. Is $\sigma(T)=\sigma{(\overline{T^{*}})}$ true for $T$? $\textbf{TRY-}$ I have proved it is true for normal operator but could not do it for bounded ...
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3answers
44 views

Product of two positive compact, self adjoint operators

If we have two positive compact , self adjoint operators; $A$, $B$. Is the product $AB$ a positive operator?
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3answers
212 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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0answers
18 views

Product of self adjoint transformations

If $A$ and $B$ are linear transformations such that $A$ and $AB$ are self-adjoint and such that $\ker (A) \subset \ker (B)$, then does there always exist a self-adjoint transformation $C$ such that ...
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61 views

Adjoint Operator of a Compact Operator

In the proof of the fact that the ad-joint operator $T^*$ of a compact operator $T$ defined on a separable, infinite dimensional Hilbert space $\mathcal H$ is also compact, I read that "$\|P_nT - T\| ...