Decomposition Definition and 414 Threads

Hello, I have a 2\times 2 real matrix M such that: M=A^T \Sigma A, where the matrix \Sigma is symmetric positive definite, and A is an arbitrary 2x2 nonsingular matrix. Both A and ∑ are unknown, and I only know the entries of the matrix M itself. Note that M is symmetric positive definite too...

Hi there, I am reading about inflation, perturbations and so on, and every book I read take the SVT decomposition as granted. Something like "every perturbation can be decomposed in Scalar, Vector and Tensor perturbations". I have 2 questions: 1) What is the definition of a Scalar, Vector and...

Okay so the partial fraction decomposition theorem is that if f(z) is a rational function, f(z)=sum of the principal parts of a laurent expansion of f(z) about each root. I'm working through an example in my book, I am fine to follow it. (method 1 below) But instinctively , I would have...

Homework Statement use partial fraction decomposition to re-write 1/(s2(s2+4) The Attempt at a Solution I thought it would break down into (A/s) + (B/s2) + ((cx+d)/(s2+4) but it doesn't.

Could you explain me: what the difference is between singular value decomposition and eigenvalue problem, when square matrices are involved. Thanks

When I'm evaluating a problem like \int \frac{2x^2 + 8x + 9}{(x^2 + 2x + 5)(x + 2)} = \frac{Ax + B}{x^2 + 2x + 5} + \frac{C}{x+2} I understand how to get the C part, that's simple. But what is a Good trick to know that I need to have Ax + B over the x^2 + 2x + 5 denominator? Is there a way I...

Just came across LU decomposition and I am not sure how to work on this problem: Let L and L1 be invertible lower triangular matrices, and let U and U1 be invertible upper triangular matrices. Show that LU=L1U1 if and only if there exists an invertible diagonal matrix D such that L1=LD and...

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an orthonormal sequence consisting of the eigenvectors of $|T|=(T^*T)^{0.5}$, (${\lambda}_{n}$)...

Homework Statement Derive x/(1-x) Homework Equations By substitution: u = (1-x) du = -dx ∫x/(1-x)dx = -∫(1-u)/u du = ∫1-(1/u)du = u-log(u)+c = (1-x) - log(1-x) + c By decomposition: x/(1-x) = 1/(1-x)-1 ∫1/(1-x)-1dx = -log(1-x)-x+c The Attempt at a Solution Which solutions...

Homework Statement ∫(2x3-4x-8)/(x2-x)(x2+4) dx Homework Equations The Attempt at a Solution ∫(2x3-4x-8)/x(x-1)(x2+4) dx Next I left off the integral sign so I could do the partial fractions: 2x3-4x-8=(A/x)+(B/(x-1))+((Cx+D)/(x2+4))...

Ok I'm stuck I have \int \frac{x^2 - 5x + 16}{(2x + 1)(x - 2)^2} \, dx and I got to this part: x^2 - 5x + 16 = A(x - 2)^2 + B(x - 2)(x + \frac{1}{2}) + c(x + \frac{1}{2})So do i need to distribute all of these and factor out or is there a simpler way? I found a solution where they are just...

Quick question... I know that if the numerator is greater than the denominator I need to divide out by long division BUT If the numerator is equal to the denominator (the exponent is what I'm talking about to be specific) then, do I need to do anything? Because I'm stuck on this problem \int...

Homework Statement (t4+9)/(t4+9t2) Homework Equations The Attempt at a Solution I'm not completely sure if I'm using the correct method to solve this. Since the degrees of the numerator and denominator are the same, wouldn't you divide the denominator into the numerator? Here is...

Hello, here goes the problem: The mass of calcium carbonate upon thermal decomposition decreased by 1/5. (a) How many molecules of CaCO3 per 100 molecules were decomposed to CaO and CO2. (b) The content of CaO in the final sample express in molar fraction. My solution: I set the starting...

In general, we heat CaCO3 to temperature of approximately 825°C it decomposes into calcium oxide and liberates carbon dioxide gas: CaCO3 →825°C→ CaO + CO2 Is it possible to heat the calcium carbonate at temperatures below 825 degrees Celsius, so the decomposition into calcium oxide.

Q3.) Express as partial fractions. a) \frac{3x+4}{x^2+3x+2} b) \frac{5x^2+5x+8}{(x+2)\left(x^2+2 \right)} c) \frac{x^2+15x+21}{(x+2)^2(x-3)}

Hi, I was wondering when you need to write the decomposition of a substance, how do you know if the number is going to be a coefficient or subscript ? Example: 2H2O ===>2H2 + 02 would be the answer But why not 2H2O ==> 2H2 + 2O Knowing that we have 2 moles of O in the beginning...

Here is the question: I have posted a link there so the OP can view my work.

I have came up with a theory, it could be completely wrong or not but i had the idea of ferromagnetic decomposition and what i mean by this is breaking a ferromagnetic object apart (such as iron) using magnetism. Is this possible or not?

Given a matrix $$A$$. Is it possible to have a Jordan block form like: $$\begin{pmatrix} \lambda & 1 & 0 & 0\\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 1\\ 0 & 0 & 0 & \lambda\\ \end{pmatrix}$$ ?

Homework Statement I need to understand how I would go about using QR decomposition of a matrix to find the matrix's eigenvalues. I know how to find the factorization, just stuck on how I would use that factorization to find the eigenvalues. Homework Equations A=QR where Q is an...

I would like to learn bit more about matrices and their decomposition. Let ##\mathbf C## be symmetric real-valued square matrix. Let ##\mathbf R## be such that $$ \mathbf R\mathbf R^T = \mathbf C. $$ Is the matrix ##\mathbf R## necessarily lower triangular (I suspect not)? Cholesky...

Homework Statement Let μ be the counting measure and m be the Lebesgue measure. Then show that on the interval [0,1] m has no Lebesgue decomposition with respect to μ. Homework Equations If such a decomposition exists, then the following holds true where X is the whole space, E is a subset...

Homework Statement Consider the n x n matrix A = diag[1,3,1] and vector x: (1,2,3) Determine the number of operations needed to compute the LU decomposition of this n x n matrix. The Attempt at a Solution So for a general n x n matrix, my prof's notes say that LU decomposition...

Do these terms practically refer to the same thing? Like a matrix is diagonalizable iff it can be expressed in the form A=PDP^{-1}, where A is n×n matrix, P is an invertible n×n matrix, and D is a diagonal matrix Now, this relationship between the eigenvalues/eigenvectors is sometimes...

I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. **Problem statement:** Let $G$ be the group of rotational symmetries of a cube, let $G_v, G_e, G_f$ be the stabilizers of a...

Hey guys, I have a problem where I am supposed to prove that R is nonsingular iff A is of full column rank in a QR decomposition. I feel like I fully understand the two major processes for obtaining a QR decomposition (Gram-Schimdt and Householder Transformations), however, I am not entirely...

Homework Statement I performed an experiment recently, which required me to heat a flask and an unknown compound to constant weight. The unknown was supposed to be a mixture of calcium sulfate dihydrate and NaCl. My goal for this lab was to determine the % NaCl in the mixture. My percent error...

Homework Statement I have a linear transformation T defined by $$ T(v_{1})=v_{1}+iv_{2}\\ T(v_{2})=-iv_{1}+v_{2}\\ $$ and I want to find a triangular matrix B of T and an invertible matrix S such that SB=AS where A is the matrix of T with respect to the basis ##\{v_{1},v_{2}\}##.The Attempt at...

My professor asks us to solve the integral of: [x/(x^4 + 1)]dx This expression is not factorable; what should I do? She is asking us to solve specifically using PFD, not u-substitution.

Homework Statement compute the LU decomposition of the 3x3 matrix: A= 2, 1, 1/2 1/2, 2, 1 1, 1/2, 2 Let f be the vector (5,3,6) Solve Lg=f and then Ux=g. You can check your answer with A_{3}x=f The Attempt at a Solution I finished the calculations: U = 1, 1/2, 1/4...

Homework Statement Fidn the total multiplications and divisions needed for the LU Decomp. of a general n x n matrix A, whose entries satisfy a_{ij} = 0 if j ≤ i - 2 Assume n=5. Also f ind the total multiplications and divisions needed for solving the lower triangular system Lg=f and for...

if there is something like (x^2+3x+6) in the denominator for one of the terms in a partial fraction problem, why do we put Ax+B instead of just A? and if the denominator is (x^2+3x+6)^2, why do we do {(Ax+B)/(x^2+3x+6)}+{(Cx+D)/(x^2+3x+6)^2}? i was always just told to memorize it, but why do we...

Hello I am stuck on an ODE involving substitution. I have done the correct substitutions, but have become stuck on decomposing the fraction. i have the following ∫(1/x)dx + ∫(u+1)/(u^2+1)du = 0 Im stuck on breaking the u down into a partial decomposition. Could anyone offer some advice on...

Hi Suppose that A \in \mathbb{R}^{3 \times 3} who maps the unit sphere in \mathbb{R}^3 to an ellips with the following semi-axes; x = \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}},0\right)^{T} \mapsto Ax = (2,0,0)^{T} x=\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}},0\right)^{T} \mapsto Ax =...

Homework Statement Find the solution of the givien initial value problem and draw its graph y''+2y'+2y = δ(t-π) y(0) = 1, y'(0) = 0 Homework Equations A Laplace transform chart would be very useful The Attempt at a Solution I chose to solve the equation with Laplace...

Hi, I hope i have posted this in the right forum I am requesting to get a supervisor who is willing to supervise me in the areas of differential equations hence get an admission for Ph.D overseas I did my M.sc project in ADM METHOD IN SOLVING THE STURM LIUVILLE EIGENVALUE PROBLEM(see...

Homework Statement The question asks one to derive the acceleration vector, $$\vec{a} = [\frac{d^2r}{dt^2} - r(\frac{d\Theta}{dt})^2]\vec{u}_r + [\frac{1}{r}\frac{d}{dt}(r^2\frac{d\Theta }{dt})]\vec{u}_\Theta$$ from the velocity vector. $$\vec{v} = \vec{u}_r\frac{dr}{dt} +...

A basis for any 3-dimensional vector space must have 3 vectors in it. So the acceleration of any object in ℝ^{3} can be decomposed into the standard basis vectors for ℝ^{3}. However, I have seen another decomposition, namely, into the tangential and normal (centripetal) acceleration vectors...

Homework Statement Give the partial fraction decomposition of 1/z4+z2 Homework Equations The Attempt at a Solution My question is about the final answer. The book gives the answer to be 1/z2+ 1/2i(z+i)- 1/2i(z-i). For my answer I keep getting a negative for both of the 1/2i...

Here is the question: Here is a link to the question: Decompose the equation into two simpler fractions? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Given the reaction: 2LiHCO_3 + SiO_2 \rightarrow Li_{2}CO_3 + SiO_2 + CO_2 + H_{2}O where SiO_2 is unaffected, the mass of the 2LiHCO_3 + SiO_2 is 9.62 g. and Li_{2}CO_3 + SiO_2 is 6.85 g., find: a) mass loss due to CO_2 + H_{2}O b) mass of LiHCO_3 in the original...

Hi all, I realize there might not be a "best method" but I want to ask if anyone has any improvements to the method taught in my class. I've looked at the Wikipedia page on this already. Let's use the matrix \left( \begin{array}{ccc} 2 & -1 & 2 \\ -6 & 0 & -2 \\ 8 & -1 & 5 \end{array} \right)...

Hi All, I would like to know how can I call or express the following process! I use a (3x3) 2D FIR Filter for imaging processing with DC = 0, like this: 0 1 1 2 O 2 /8 1 1 0 My filter is such that I can decompose it into finite sates, as my image (medical) can take 9...

Hi, I finished reading about the Schmidt decomposition from Preskill's notes today. I understand and follow his derivation but it still seems completely non intuitive to me. We have \mid\psi\rangle_{AB}=\sum_{i,u}a_{iu}\mid i\rangle_{A}\mid u\rangle_{B}=\sum_{i}\mid...

Here is the question: Here is a link to the question: Help with Calculus BC: partial fractions!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Consider a quantum system with a countable number of basic states \left|n\right\rangle. Calculate the decomposition into a basis of coherent states \left|λ \right\rangle all obeying \hat{a} \left|λ \right\rangle = λ \left|λ \right\rangle Homework Equations \hat{a}...

I am trying to determine the change of heat for: N2H4(g) -> N2(g) + 2H2(g) Here is what I did and what I got the correct answer is -86.0 which I am clearly not getting [(163)+ (2*436)] - [(4*391) + (163)] = -692 thanks for any help!

Hey, I have this code for lu decomposition but It doesn't quite work. If anyone could help me with the problem I'd be very appreciative. for(j=0; j<N; j++) for(i=j+1; i<N; i++) U[i][j]=0; for(j=0; j<N; j++) for(i=j+1; i<N; i++)...

What is the best way of introducing singular value decomposition (SVD) on a linear algebra course? Why is it so important? Are there any applications which have a real impact?