nonlinear optimization – Confusion about the optimal parameter value non linear least squares

The normal equation for the nonlinear least squares is denoted as

$ boldsymbol{Delta} boldsymbol{beta}=left(mathbf{J}^{mathrm{T}} mathbf{J}right)^{-1}mathbf{J}^{mathbf{T}} boldsymbol{Delta} mathbf{y}$

Where

$beta_{j} approx beta_{j}^{k+1}=beta_{j}^{k}+Delta beta_{j}$

In order to perform the nls I have to predetermine starting values of $beta$.

However, once I have done that and calculate $Deltabeta$ how do I obtain the optimal value for the parameter?

I do not fully understand the GaussNewton Method. Could anyone please explain the procedure for finding the optimal parameter?

linear algebra – Finding which members of a family of (possibly infinite-dimensional) matrices have trivial null space

Background

I have a set $S$ (that is possibly infinite) and a correspondence between functions $c:S^3tomathbb{R}$ (I will write $c(i,j,k)$ as $c_{ijk}$) and matrices $M$ with rows indexed by $(i,j,k)in S^3$ and columns indexed by $(m,n)in S^2$ defined as follows:

$$M_{(i,j,k),(m,n)}=delta_{im}c_{njk}+delta_{jm}c_{nik}-delta_{ln}c_{ijm}$$

where $delta_{ij}$ is the kronecker delta function.

Question

I want to determine some necessary and sufficient conditions on $c$ so that the above matrix has trivial null space.

Any references, ideas, techniques, etc. would be appreciated. Of course, I’m not looking for a complete solution to the problem, I just don’t know what sort of methods might even be used to attack this (despite it appearing at first to be a simple linear algebra problem).

Progress

For finite $S$ with cardinality $N$, this is a $N^3times N^2$ size matrix, so it should be true that most choices of $c$ result in a matrix with trivial null space. However, I’m still not sure what “most” would even mean in this context.

I’ve also proven the result for $c_{ijk}=delta_{ijk}$ (that is $c_{iii}=1$ and equals zero otherwise), but otherwise have no further results.

linear algebra – Visualize the geometric interpretation of the matrix power of a matrix with complex eigenvalues

I can understand the geometric meaning of $A^n$ (here $A in R^{n times n}$) when the eigenvalues of $A$ are all real. Basically, you scale up the each eigenvector $v_i$ along its direction by $lambda_i$.

But what happens if the eigenvalues are complex? In that case, the eigenvectors will have complex elements too. I think I can sort to guess, $lambda_i$ here does a bit of the rotation. But how do I define the direction of an eigenvector, with complex elements?

linear algebra – Using dimension formula to prove subspace dimension formula

I’ve always suspected the formulas $dim F+G = dim F + dim G – dim Fcap G$ and $dim V = dim mathrm{Ker} T + dim mathrm{Im} T$ were related someway. So i tried proving the former using the latter.

It is easy to see that the direct outer product $Ftimes G$ has dimension the sum of dimensions of $F$ and $G$. Define $pi: Ftimes G twoheadrightarrow F+G$ by $pi(f,g) = f+g$. It is clearly surjective. I claim that the kernel of $pi$ is the subspace $H$ of $(Fcap G)^2$ of elements of form $(f, -f)$, and therefore of dimension equal to the one of $Fcap G$. Therefore the former result results from applying the dimension formula to $pi$.

Is this correct?

linear algebra – Demonstrating that vectors S1 and S2 are base of R3 and their base change matrix

Could someone explain and show me how you would demonstrate that the vectors S1 ={(1,1,1),(2,4,3),(-3,2,3)}. and S2 ={(1,2,1),(-1,-1,0),(2,9,8)}. are base of R3.

Secondly how would you determine de base change matrix from B1 to B2.

Thank you really much.

Operators "building" linear independant sets

Let $E$ be a separable Banach space and let $Tin L(E,E)$.

Is there a condition on $T$ ensuring that:
$$
mbox{${x_n}_{n=1}^Nsubseteq E$ is linearly independent} Rightarrow
{T(x_n)}_{n=1}^Ncup {x_n}_{n=1}^N mbox{ is a independent in $E$}?
$$

Is $T$ being chaotic or mixing enough for this?

A question about Functional Analysis : The linear operator is surjective

The Question says:

Let be $varphi: X rightarrow mathbb{C}$ linear. If $varphi$ isn’t null then $varphi$ is surjectvive.

I have no idea how to do this.
The only thing that I know is that ‘ how $varphi$ isn’t null, then exists a $xi in X$ such that $varphi (xi ) = a+ i b neq 0. $
But I can’t see how this implies that for all $a+ib in mathbb{C}$ exists $ x in X$ satisfying $varphi (x ) = a+ i b. $

nonlinear – System of Non Linear Equations

I have a system with 7 equations and 7 unknowns with 11 parameters. I wish to get closed form solutions for each of those 7 equations. However, the system is non linear in the parameters which makes the solution even more complicated. Is there any way to solve this? Or should I attempt to linearize the system of equations? If linearizing is the option how should I do it in Mathematica?

linear algebra – Show T is an isomorphism

Define $T:V→ W$ by $T( a₁ +a₂t +a₃ t²+a₄ t³ ) =left(begin{array}{l}frac{1}{√{1} }a₁ & frac{1}{√{2} }a₂ \frac{1}{√{4} }a₄ &frac{1}{√ {3} }a₃ end{array}right)$ . Show that T is an isomorphism such that for any $p( t ) ,q( t ) ∈ V ⟨ Tp( t ) ,Tq ( t ) ⟩_{w}=⟨ p( t ) ,q( t ) ⟩_v$ . $Where ⟨ , ⟩ _{w}$ is the Frobinious inner product on W , i.e., $⟨ A,B ⟩ _{w} = tr( A^TB )$.

This is a homework given to us and I know that a linear transformation is isomorphic if its kernel only contains the zero vector. I’m not entirely sure how to proceed with this specific problem.

linear algebra – Computing the exponential of a $2 times 2$ matrix using trace $0$ matrices

It is an easily proved fact that for a $2times 2$ traceless matrix $A$,

$$ e^A = cosleft(sqrt{det(A)}right)I + frac{sinleft(sqrt{det(A)}right)}{sqrt{det(A)}}A$$

Problem 2.7 of Lie Groups, Lie Algebras, and Representations by Bryan Hall asks to use this fact to compute $exp(X)$, where

$$ X = begin{pmatrix}
4 & 3\
-1 & 2
end{pmatrix}$$

In other words, I have to write $X$ in terms of traceless matrices, and employ the above fact. My question is: is there a systematic way to do this?

My idea to solve this problem is to write $X = X_1 + X_2$, where $X_1$ is traceless, $X_2$ is diagonal or nilpotent, and $(X_1, X_2) = 0$, and compute the exponent using $e^{X_1 + X_2} = e^{X_1}e^{X_2}$. For example, I tried the most obvious thing:
$$X = begin{pmatrix} -2 & 3\-1 & 2end{pmatrix} + begin{pmatrix} 6 & 0\0 & 0end{pmatrix},$$

but the two matrices above do not commute.