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Questions tagged [canonical-transformation]

Use this tag for questions related to canonical transformations, which are changes of canonical coordinates that preserve the form of Hamilton's equations.

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Minimal polynomial for any power of Jordan block is same as the minimal polynomial of the Jordan block.

Let $J$ be the $n \times n$ Jordan block corresponding to the eigen value $1$. For any natural number $r$ is it true that the minimal polynomial for $J^r$ is $(X-1)^n$ ? Another way to think about it ...
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$u_{xx} + u_{xy} + u_{yy} = 0$ in canonical form

How do i put $u_{xx} + u_{xy} + u_{yy} = 0$ in canonical form? $a=1, b=1/2, c=1 $ implies that it is elliptic as $b^2 - ac <0$ $dy/dx = \lambda$ where $a\lambda^2-2b\lambda+c=0$ gives $\lambda = \...
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Fixing the bounds of an indefinite integral

Question: $$\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial t^2}=f(x,t) \qquad \qquad u(x,0)=\frac{\partial u}{\partial t}(x,0)=0$$ Solve this equation, writing the solution in the ...
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Adaptive Pole Placement and Canonical forms

I have a linear system in the observable canonical form as \begin{align} \dot{x} &= \underbrace{\begin{pmatrix} -a_2 & 1&0\\ -a_1 & 0&1\\ -a_0 & 0&0 \end{pmatrix}...
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Families of step-wise canonical transformations.

Do there exist "families" of step-wise canonical transformations, in the sense that: $${\bf A = TCT}^{-1}, \\{\bf T = T_2}^n$$ So that $${\bf T_2}^k {\bf C} {\bf T_2}^{-k} $$ Also has some canonical ...
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Canonical isomorphism between $\Lambda^2 V$ and $\mathbb{R}$

Let us take a two-dimensional real vector space $V$, and define $$\det \colon V \times V \rightarrow \mathbb{R} \colon (u,v) \mapsto \det(u\vert v).$$ Since this map is bilinear, the universal ...
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Linear programming positive constraint conversion

So lets suppose we have the following linear problem: $\min -x+2y$ $s.t$ $x-y\le6$ $x-3y\le12$ $x\ge0 ,y\le2$ and I want to use simplex to find the optimal solution. First we convert this ...
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Trying to understand hyperbolic canonical form transformation

I can't figure out what this author did in the text pictured below. Can someone PLEASE help. The author is transforming hyperbolic PDE's into canonical form. The author displays a general 2nd order ...
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PDE $au_{xx} +2bu_{xy}+cu_{yy}$ into canonical form [closed]

I am looking for the right substitution, to transform the pde $au_{xx} +2bu_{xy}+cu_{yy}=f$ with $ac \gt b^2$ into the canonical form $\nabla^2U(\hat x,\hat y)=F(\hat x,\hat y)$ where $\nabla^2$ is ...
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Typo in Spivak’s Mechanics?

In the last string of equalities, shouldn’t $ f^*(dq + \tau dp)\wedge f^*dp $ instead read $(dq + \tau dp)\wedge dp$? $Q, P, q, p$ are coordinate functions on $T^*M$, and $f:T^*M \rightarrow T^*M$. ...
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Canonically isomorphic

I am currently working through the exercises in 'Principles of Harmonic Analysis' For a closed subgroup B of the LCA-group A and a closed subgroup L of $\widehat{A}$. Let $B^{\bot}=\{\chi\in\...
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Reduce the equation to canonical form. $u_{xx}+2ayu_{xy}+e^{2x}u_{yy}-u=0$

Consider the equation $$u_{xx}+2ayu_{xy}+e^{2x}u_{yy}-u=0$$ where $a$ is a real constant. Determine the value(s) of $a$ when this equation is elliptic everywhere in the $xy$-plane in which case find ...
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Canonical equations (PDE)

Consider the equation $$u_{xx} +2ay u_{xy} +e^{2x} u_{yy} −u = 0,$$ where $a$ is a real constant. Determine the value(s) of a when this equation is elliptic everywhere in the $xy$-plane in which case ...
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Find a General solution to the equation, leading him to the canonical form.

enter image description hereHere it is my question. uxx + 10uxy + 25uyy + ux + 5uy = 0 I recognize that it is a hyperbolic PDE. . I don't know how to proceed further to get the canonical form. can ...
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Deriving the equations defining a symplectic map

The objective is to derive $$p_k=\frac{\partial\phi}{\partial q_k},\quad Q_k=\frac{\partial\phi}{\partial P_k},\quad \tilde{H}=H+\frac{\partial\phi}{\partial t},$$ where $\phi=\phi(t,q,P)$. I ...
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How do I find canonical coordinates for the Lorentz group generators?

Consider the Poisson brackets (symplectic structure) given by the Lorentz algebra (Lie algebra of $SO(1,3)$) $$\{M^{AB},M^{CD}\} \equiv \omega_{AB,CD}\mathrm{d}M^{AB} \mathrm{d} M^{CD} = \eta^{AC} M^{...
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What is the matrix of $f$ in the canonical bases of the two spaces?

Question: Consider a linear map $f∈L(Π^2(\Bbb R),\Bbb R^2)$ whose matrix $f_{G,B}$ in the basis $B$ of $Π^2(\Bbb R)$ . $G=\{{ <1,-1>^T , <2,1>^T \}}$ of $\Bbb R^2$ is: $f_{G,B}= [<−10,−...
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Linear Transformation $T(f(x)) = f(0)x-f'(1)$ for $P_3(\mathbb{R})$

For the standard basis $\{1, x, x^2, x^3\}$, I get the following matrix: $$ A= \begin{pmatrix} 0 & -1 & -2 & -3 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & ...
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Process of finding the canonical form of conic section

I have a problem with understanding the process of transforming a conic section into canonical form. My conic is: $11x^2-24xy+4y^2+2x+16y-11=0$ Can someone explain me the complete process?
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Injection from dual space into double dual space

I know for a vector space $V$ over the real or complex numbers, there exists a canonical embedding into its double dual $V^{**}$, and if $V$ is given an inner product, then there is a canonical ...
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variations of the canonical form of a second order PDE

Is it true to say that for each PDE of second order, there is one single canonical form? (or at least to the extant of multiplication by a constant/function of the original variables). So far it it ...
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Why is this the general solution to this system of linear partial differential equations?

For a system of PDE's given by $$ \begin{cases} \frac{\partial u}{\partial x} + 6 \frac{\partial u}{\partial y} = 0 \\ \frac{\partial^2 u}{\partial x^2} - 6 \frac{\partial^2 u}{\partial y^2} = 0 \end{...
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How do you solve a system of PDEs in canonical form?

Suppose you wish to solve a system of $n$ PDEs, which can be expressed in vector form as $$ A \boldsymbol{u}_x + B \boldsymbol{u}_y = \boldsymbol{c} \hspace{10mm} (*_1) $$ where $$ A = \begin{pmatrix}...
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Matrix calculus derivative form to canonical differential form

On https://en.wikipedia.org/wiki/Matrix_calculus under Conversion from differential to derivative form we can see the equivalence: How can this be proven ?
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MIMO state space <--> MIMO transfer function - How?

Normally I use the controllability and observability canonical forms to transform a transfer function into a state space model. I also find the poles, zeros and gain from a state space model to ...
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1answer
359 views

Conversion from differential to derivative for trace of a matrix

I am studying the matrix calculus page written in wikipedia and I have a question. In the table ' $\text{Identities: scalar-by-matrix}\frac{\partial y}{\partial \mathbf{X}}$ ' , it is shown : (1) $\...
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Observable canonical form - Transfer function to state space

Let's say that I have a transfer function: $$G(s) = \frac{0s^4 + 3s^3 + 2s^2 - 2s - 4}{10s^4 + 2s^3 + s^2 -0s + 10}$$ And I get the state space model by using observable canonical form: $$ A = \...
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Linear algebra and geometry

I can't figure out how to get to the answer of this question, been staring at it for hours Let $$q(x_1, x_2, x_3) = x_1x_2 + x_1x_3 + x_2x_3$$ be a quadratic form in three unknown variables over $\...
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1answer
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Jordan Canonical Form set of polynomials over complex field

N: $P_4(C) \to P_4(C)$ defined by $N(p) = p''- 3p'$ Find the canonical form and a canonical basis for the mapping of $N$. I am not sure how to compute this when it is over $\mathbb{C}$ instead of ...
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Find a matrix which is a Jordan canonical form for A

Suppose A is an element of $M_{7x7} \mathbb{C}$ is a matrix with characteristic polynomial: x(A(t))=$(2-t)^2(3-t)^2(4-t)^3$ Suppose as well that:dim}N(A-2I) = 1 , dim N(A-3I) = 2,dim N(A-4I) ...
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1answer
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Finding Canonical form using given functionals

Let $V,W$ be vector spaces over field $\mathbb F$. Let there be two functionals; $\psi : V \to \mathbb F$ , $\phi : W \to \mathbb F$. How to find the Canonical form (of orthogonal matrix, meaning it ...
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1answer
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Estabilish wheter there exists a basis of $R^3$ made of eigenvectors of P or not, and if so, wheter such a can be made of mutually orthogonal vectors

$Pv=v-(w\cdot v)\cdot w$ Where $w=(1,−1,0)$ i.Write the matrix $A$ that represnts $P$ in the canonical bases of $\mathbb{R}^3$ ii.Find the eigenvalues and eigenvectors of $P$ iii. Estabilish ...
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Find a basis consisting of a union of disjoint cycles of generalized eigenvectors

$T$ is the linear operator on $P_2(\mathbb{R})$ defined by $T(f(x))=2f(x)-f'(x)$ 1) Find a basis for a generalized eigenspace of $T$ consisting of a union of disjoint cycles of generalized ...
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Canonical rotation to take a unit vector to another [duplicate]

In 3 (and 2 for that matter) dimensions, there is a "canonical" (independent of arbitrary choices) rotation that takes a given unit vector $u$ to another unit vector $v$ (edit: when $u,v$ are in ...
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Which matrices are permutation diagonalizable?

Are there some results for what must hold for a matrix of complex (or real) entries $\bf M$ for it to be similar to a permuted diagonal matrix $\bf D$: $${\bf S(PD)S}^{-1} = \bf M$$ where $\bf P$ is ...
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Block eigenvalue decompositions with strictly positive elements.

In the similarity transformation $$\cases{{\bf A = T}^{-1}{\bf DT}\\ {\bf D}_{ij} = \lambda_j\delta_{i-j}}$$ The eigenvalues $\lambda_j$ of $\bf A$ are on the diagonal in $\bf D$, and corresponding ...
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Some sort of rank-aware determinant? Would it be useful?

Assume $\bf A$ has a Jordan form: $${\bf A = SJS}^{-1}$$ If any diagonal value of $\bf J$ is $0$ then $\bf A$ does not have full rank and therefore $\det({\bf A}) = 0$. Claim: We can define a ...
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Find an integer quartic root of a $3\times 3$ matrix

Find a $3 \times 3 $ matrix $X$ with integer coefficients such that \begin{align*} X^{4} &= 3 \begin{bmatrix} 2 &-1 &-1 \\ -1 &2 &-1 \\ -1 &-1 &2 \end{bmatrix}. \end{...
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Markov Chain expected number of visits

\begin{pmatrix} 0&0&.2&.8&0\\ 0&0&0&.9&.1\\ .6&0&0&0&.4\\ .2&.8&0&0&0\\ 0&.9&.1&0&0 \end{pmatrix} Question: Suppose ...
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Conditions for controllability canonical form to be non-observable but detectable

I have a system in controllability canonical form: $\dot{x} = \begin{bmatrix} 0 & 1 & 0 & ... & 0 \\ 0 & 0 & 1 & ... & 0 \\ . & . & . & ... & . ...
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Group of permutations

I have a problem $\mathcal{P}(s_1,\dots,s_5)$, $s_i\in\mathbb{N}$, which is invariant by circular permutations ($\mathcal{P}(s_1,s_2,s_3,s_4,s_5)=\mathcal{P}(s_2,s_3,s_4,s_5,s_1)=\dots$) symmetry: $\...
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Canonical transformation of a Hamiltonian system

Consider the following Hamiltonian: \begin{equation} H_{2}(q,p) = \dfrac{p_{x}^{2}}{2} +\dfrac{p_{y}^{2}}{2}+\dfrac{p_{z}^{2}}{2}-xp_{y}-x^{2}-\dfrac{x^{2}}{x_{s}^{3}}-x_{s}^{2}+p_{x}y +\dfrac{y^{...
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Basis-free formula for $\mathrm{Hom}_k(V,V)\rightarrow V^*\otimes V$

Let $V$ be a finite dimensional vector space over a field $k$. Then there is a natural map $\phi:V^*\otimes V\rightarrow \mathrm{Hom}_k(V,V)$ given by $$\phi:f\otimes v\mapsto \Big(x\mapsto f(x)v\...
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Canonical forms which have “minimal” Gershgórin discs, do they exist?

I'm wondering about if there is some way to define, uniquely or not a canonical form which has minimal radii for Gersgórin discs. To be more specific for a given matrix $\bf A$, find $\bf C$ and $\bf ...
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1answer
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Jordan Canonical Form of complicated Matrix

$$B:= \left[ \begin{matrix} -1 & 9 &0 &0 &0 \\ 0 & -1 & 0 & 0 & 0 \\ 0&3&-1&0&0 \\ 0 & 1 & 1 & 1 &...
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How to list all possible dimension of $\ker{T},\ker{T^2},…,\ker{T^{k-1}}$ and the corresponding canonical forms?

Let $V$ be $5$-dimension vectorspace, and $T:\ V\rightarrow V$ a nilpotent linear transofrmation of order (index) $k$ where $1\le k\le 5$. How to list all possible dimension of $\ker{T},\ker{T^2},...,\...
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1answer
772 views

Finding all the possible dimensions of Kernel with a nilpotent linear transformation

Let $V$ be a 5-dimensional vector space, and $T : V → V$ a nilpotent linear transformation of order $k$, where $1 ≤ k ≤ 5$. Make a list of all the possible dimensions of $\ker(T), \ker(T^2), \ldots , \...
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1answer
136 views

Canonical form transformation

​ ​ My subject about the canonical form of PDE. I had many exercises to do and they were fine, but I'm stuck with this one: ​ ​ $$U_{xx}-yU_{xy}+xU_x+yU_y+u=0$$​ ​ So first we have to calculate $B-4AC=...
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1answer
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Canonical form simplex method

In 2-phases simplex method what kind of operations must be done to get the canonical form tableau? In this step(phase 2 of 2-phases method) after the remotion of artificial variables columns of ...