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Questions tagged [adjoint-operators]

For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).

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61 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 &...
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292 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
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1answer
101 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 ...
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1answer
108 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 ...
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3answers
133 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?
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1answer
48 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?
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2answers
441 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 ...
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1answer
40 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? ...
3
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1answer
126 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 ...
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3answers
698 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 ...
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1answer
63 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} = x^tA^t\overline{...
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1answer
78 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: (...
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2answers
150 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 ...
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1answer
139 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 ...
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1answer
336 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 = ...
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0answers
169 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 ...
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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 = \...
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1answer
2k views

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

What is the adjoint of an inverse matrix? Is $(T^{-1})^{*} = (T^{*})^{-1}$?
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2answers
488 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 $x,y \in H$ $4\...
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3answers
750 views

Proof that All Entries in an Inverse Matrix are Integers

"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 {...
3
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1answer
472 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 ...
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2answers
433 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 ...
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1answer
2k 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 ...
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2answers
10k views

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

So, $A$ is a $n \times n$ matrix with integer entries. 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 ...
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1answer
36 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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1answer
29 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
341 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? ...
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1answer
293 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
49 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} \Phi(x)=\frac{1}{2}(Tx,x)...
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2answers
167 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
85 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. $\langle f, B f \rangle \geq 0 \forall f \in D(B)$. ...
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1answer
649 views

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

I know how to find the adjoint of $2\times 2$ matrix but I'm at a loss for finding a general $2\times 2$ matrix where $A = adj(A)$. Thanks for your help!
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4answers
120 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 ...
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1answer
191 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}, \boldsymbol{w}...
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1answer
1k 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 ...
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2answers
186 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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1answer
47 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.. :/
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3answers
526 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 ...
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1answer
95 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 $A^*A$. ...
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1answer
79 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} \right)...
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0answers
428 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 spaces....
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0answers
37 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?
2
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1answer
451 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 ...
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2answers
706 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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1answer
48 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
51 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 T^*(x),y\...
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1answer
3k 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: $[im(T)]...
2
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1answer
689 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 ...
4
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1answer
685 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 that ...
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2answers
5k 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) = (AdjA)^{...