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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Show $\mathrm{rank}\mathsf{T} = \mathrm{rank}\mathsf{T}^\ast$ for a linear operator of finite-dimensional inner product space

I need to show that $\mathrm{rank}\mathsf{T} = \mathrm{rank}\mathsf{T}^\ast$ for a linear operator $\mathsf{T}$ on a finite-dimensional inner product space $\mathsf{V}$. Let $\beta = \{v_1,\dots,v_n\...
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2 answers
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Finding Matrix adjoint

Need someone to check my reasoning as I don't feel confident in this topic: consider $P_2$(C) with inner product $$<p(x), q(x)> = \int {q(x)p(x) dx} $$ T is defied by T(p(x)) = p'(x) + p(x) ...
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Regular Sturm-Liouville Boundary Value Problem

Let $L[y]:=y''''$. Let the domain of $L$ be the set of functions that have four continuous derivatives on $[0,π]$ and satisfy $y(0)=y'(0)=0$ and $y(π)=y'(π)=0$ a) Show that $L$ is self adjoint b) ...
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self adjoint linear operator and integration

is this formula correct ?? $$ \int_{-\infty}^{\infty} Lf(x)\delta (x-1)= \int_{-\infty}^{\infty} f(x)L^{\dagger}\delta(x-1) $$ here $ L $ is a linear operator and $ L^{\dagger}$ is its formal ...
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Showing an operator is self adjont

I am trying to show that the operator: $$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$. So I can re-write this operator ...
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2 answers
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Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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1 answer
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self-adjoint operator and unitary orthogonal matrix

Please offer a solution to the following problem. It was offered in class by my professor as an additional exercise to try on one's own. Let $V$ be the inner product space, and assume that $\alpha \...
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1 answer
42 views

Spectrum of $H + H^*$ in Hilbert space

Consider a bounded operator $H$ on a Hilbert space $V$. Let ${\rm Re\ sp} (H)$ be the real parts of all points in the spectrum of $H$. Is it ture that ${\rm sp}( H + H^*) = 2 {\rm Re\ sp}(H)?$ Or ...
-1 votes
1 answer
68 views

Finding adjoints of linear operator [closed]

I’m going through Linear Algebra Done Right, learning about singular value decomposition. I’m a bit lost when trying to calculate an adjoint of an operator. Obviously, this is simple when you are ...
-1 votes
2 answers
291 views

Matrix representation of adjoint & co-adjoint orbit of $so(3)$

So I am trying to find the co-adjoint orbits of the lie algebra $so(3)^*$ from this example but I am stuck with a very trivial linear algebra property now I found the adjoint orbits and I know the ...
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Describe the closure and adjont of operator T

Let $\mathcal{H}_2$ be a separable Hilbert space and $(T,D(T))$ be an operator in $\mathcal{H}_2$ such that there exists a Hilbert basis $(e_n)_{n\in \mathbb{N}}$ of $\mathcal{H}_2$ with $e_n \in D(T)$...
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Show that $\text{ker}(T^*T)=\text{ker}(T)$

Show that $\text{ker}(T^*T)=\text{ker}(T)$ In my solutions when showing that $x\in\text{ker}(T)\implies x\in\text{ker}(T^*T)$ the following is given: $$x\in\text{ker}(T)\implies Tx=0\implies T^*Tx=0\...
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1 answer
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Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open $H:=L^2(\Omega,\mathbb R^d)$ $U$ be a separable $\mathbb R$-Hilbert space $Q:U\to H$ be ...
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2 answers
456 views

Isometric <=> Left Inverse Adjoint

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ $$...
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1 answer
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How to find adjoint?

where From this question I concluded that L^T = L*. But when I find the transpose and then adjoint of the matrix I get as the adjoint L* of L. But it is wrong. What am I doing wrong?
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2 answers
142 views

If $\det(A) = 1$, what are possible values of $\det(\mbox{adj}(A))$?

If A is square matrix (3×3), and $det(A)=1$, is it true that $det(adj(A))=1, -1$?
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1 answer
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Question on self adjoint properties

I have the follwing question and have managed to do it all apart from the last 3 parts of c) showing the various statements. Cant figure out where to start! Any help will be appreciated!
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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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1 answer
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Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
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1 answer
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Prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n =0 $ iff $a_{n} \in \mathbb{R}$ [closed]

How to prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n=0 $ iff $a_{n} \in \mathbb{R}$ and $[u_n]$ is an orthonormal sequence? Edit: does it have something to do with the equality: $\...
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1 answer
84 views

Describe the closure of operator S

Let $(\lambda_n)_{n\in N}$ be an arbitrary sequence in $\mathbb{C}$ and hilbert space $\mathcal{H}_1 := l^2(\mathbb{N})$. Consider the operator $(S,D(S))$: \begin{align*} D(S) := \{x\in\...
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3 answers
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Product of two positive compact, self adjoint operators [duplicate]

If we have two positive compact , self adjoint operators; $A$, $B$. Is the product $AB$ a positive operator?
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1 answer
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Applying the definition of adjoint operator [closed]

How do I apply the definition of adjoint operator in this problem? U and V are two arbitrary operators, not necessarily Hermitian. Show that (UV )† = V †U†.
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