## SVD perturbation lower bound

Suppose I have a matrix $$X in mathbb{R}^{a times b}$$, with $$a > b$$ and $$X$$ full rank with singular value decomposition $$UDV’=X$$. Then, suppose I also have a “small” matrix $$A$$ (i.e., some perturbation matrix) which is not equal to $$X$$. I am interested if the following quantity could be lower bounded by a norm of $$A$$ and norm of $$X$$:

$$sup_{|Q| leq 1} {rm tr}left[(Q – UV’)'(X – A) right],$$

where $$|cdot|$$ is the spectral norm. Using that the dual norm of the “nuclear norm” is the spectral norm, the above quantity could also be expressed as:
$$|X – A|_* – |X|_* + {rm tr}(VU’A),$$
where $$|cdot|_*$$ is the nuclear norm which sums the singular values of its argument.

Any suggestions for directions or references would also be appreciated.

## Does a system stay balanced after a perturbation?

Let’s take the following example for concreteness:

I’m biking, and I stop at an intersection (not asking how bikes stay up while in motion). My feet are off the ground, and the center of mass (CM) of myself and the bike is perfectly over the wheels. Everything is at rest.
Now, I move my right leg outward, to the right. Taking the point of contact of the wheels with the ground to be fixed…

Do I lose balance? If yes, which way do I fall? Does it depend? On what?

Let’s say I start with the CM a little off to the left, then move my right leg out, which way do I fall?

## computation of an integral for 2nd order non degenerate perturbation theory

I am given that the potential of a diatomic molecule is equal to $$V(rho)=-2V left ( frac{1}{rho ^2}-frac{1}{2 rho ^2} right )$$ With $$rho=r/a$$ is a dimesionless coordinate, and $$r$$ is the separation distance between the two atoms. I found the first order corrections without issue, but I am stuck on finding the second order one. I know that
$$E_n^2=sum_{mneq n}frac{|langlepsi_m^0|H’|psi_n^0rangle|^2}{E_n^0-E_m^0}$$
For the given problem, I also found that the wavefunctions are $$psi=e^{-x^2/2} H_{n}(x)$$ (hermite polynomials). My problem is in computing the integral of the inner product, i.e, computing
$$langlepsi_m^0|H’|psi_n^0rangle=int_{-infty}^{infty} x^3 e^{-x^2} H_{n}(x)H_{m}(x)dx$$
I could apply integration by parts a bazillion times to obtain the answer, but it is far too tedious. From reading Griffiths, I know that there is a way to do this much simpler with dirac notation and ladder operators, in conjunction with the usual ladder operator identities, but I am unsure on how to go about this.

## SVD perturbation lower bound

Suppose I have a matrix $$X in mathbb{R}^{a times b}$$, with $$a > b$$ and $$X$$ full rank with singular value decomposition $$UDV’=X$$. Then, suppose I also have a matrix $$A$$. I am interested if the following quantity could be lower bounded by a norm of $$A$$:

$$sup_{|Q| leq 1} {rm tr}left[(Q – UV’)'(X – A) right],$$

where $$|cdot|$$ is the spectral norm. Using that the dual norm of the “nuclear norm” is the spectral norm, the above quantity could also be expressed as:
$$|X – A|_* – |X|_* + {rm tr}(VU’A),$$
where $$|cdot|_*$$ is the nuclear norm which sums the singular values of its argument.

Any suggestions for directions or references would also be appreciated.

## Perturbation theory first order

To begin with, I have a planar interface $$Z_0$$. Now I would like to introduce a harmonic pertubation $$phi(x,t) = delta(t)sin(omega x)$$ into the planar interface.

Now, my question is which expression would represent the shifted coordinates with the “first-order” perturbation correction ?

$$C(x,z) = C(z) + F(z)delta(t)sin(omega x)$$ is this correct? or $$C(x,z) = C(z) + delta(t)sin(omega x)$$

## Perturbation of Identity is isomorphism

Let $$X$$ be a Banach space and $$K:Xto X$$ be a linear, bounded operator with $$Vert KVert<1$$. Show that $$(I-K)^{-1}$$ exists and is bounded.

Let $$Vert KVert = 1-varepsilon$$. Then we get $$varepsilonVert xVert leq Vert (I-K)xVert leq (2-varepsilon)Vert xVert,$$ so $$I-K$$ is bounded and injective. If we have surjectivity, it follows that $$I-K$$ is an isomorphism since $$X$$ is Banach. While it seems clear that it should be surjective, I have so far failed to prove it.

## Substitution of a perturbation into a Partial Differential Equation

I have a second order partial differential equation.

$$frac{partial^2 U}{partial x^2} + frac{partial^2 U}{partial z^2} + frac{2}{l}frac{partial U}{partial z}=0$$.

I need to introduce a perturbation $$zeta(x,t)$$ into the above equation in order to shift the coordinates. Here, $$zeta(x,t)=hat{zeta}exp(ikcdot x+omega t)$$, where, $$k$$ is a two-dimensional wave vector and $$omega$$ the wave number.

After substitution I get the following equation :

$$-k^2hat{zeta}exp(ikcdot x + omega t)frac{partial^2U}{partial zeta^2} + frac{2}{l}frac{partial U}{partialzeta}=0$$. As, $$frac{partial^2 U}{partial x^2} = frac{partial^2 U}{partial zeta^2}frac{partial^2 zeta}{partial x^2}$$.

But, when I look at the solution for this problem it is given as,

$$U = exp(-2z/l) – 1 + hat{zeta}exp(ikcdot x+omega t – qz)$$, where $$q$$ is the solution of a quadratic equation.

So I’m stuck at the substituted equation, as it will not result in the given solution. Can anyone point out where I’m going wrong in the substitution?