rank Definition and 290 Threads

Hey! :o Let $\mathbb{K}$ be a fiels and $A\in \mathbb{K}^{p\times q}$ and $B\in \mathbb{K}^{q\times r}$. I want to show that $\text{Rank}(AB)\leq \text{Rank}(A)$ and $\text{Rank}(AB)\leq \text{Rank}(B)$. We have that every column of $AB$ is a linear combination of the columns of $A$, or not...

Homework Statement Homework Equations N/A The Attempt at a Solution I know that they got a rank of 2 since there are 2 linearly independent columns but what if we decided to count rows? In that case we would have 4 linearly independent rows which would suggest the rank is 4? How do we...

Is there any shortcut to find the rank of this $4 \times 6$ matrix quickly? $$A = \begin{pmatrix} -3 &2 &-1 &-2 &7 &-1\\ 9 &2 &27 &18 &7 &-9\\ 3 &2 &1 &0 &7 &-1\\ 6 &2 &8 &4 &-7 &-4\\ \end{pmatrix}$$ The above is a sample question for semester final test. If it were a homework, of course I...

I would love to get help on this problem: Suppose that $M$ is a square $k \times k$ matrix with entries of 1's in the main diagonal and entries of $\frac{1}{k}$ for all others. Show that the rank of $M$ is $k$. I think I should go about by contradiction, that is, by assuming that the column...

Let the matrix of partial derivatives ##\displaystyle{\frac{\partial y^{j}}{\partial y^{i}}}## be a ##p \times p## matrix, but let the rank of this matrix be less than ##p##. Does this mean that some given element of this matrix, say ##\displaystyle{\frac{\partial y^{1}}{\partial u^{2}}}##, can...

What is the rule on the preservation of rank through an elementary operation? I know that rank can never go up, but is there any direct way to determine that it goes down (either than reducing the matrix down to row-echelon form)? Is there a good source that go into the proofs for properties...

I am dealing with a 3x4 complex matrix M that relates a vector d to another vetor c. That is: c = [M]*d d is 4x1 and c is 3x1. I want to introduce a new line (constraint) into M, say d(1) = d(2). However, I would like to only apply the constraint to the real or only the imaginary parts. Is...

Homework Statement Let A,B be square matrices of order n. n>=2 lets A and B be matrices of Rank 1. What are the options of the Rank of A+B ? Homework EquationsThe Attempt at a Solution I know that there are 3 possibilities, 2, 1 , 0. Just having trouble with coming up with a formula. i tried...

I am reading in my group theory book the well known commutation relations of the Lie algebra of SO(3), i.e. [J,J]=i\epsilon J. What I don't understand is the statement that "from the relations we can infer that the algebra has rank 1". Any ideas?

Homework Statement Which bulb(s) will have the higher light intensity?Homework Equations Resistors: Connection in series: same current Connection in parallel: same delta V The Attempt at a Solution A and C are in series. They'll be equally bright. B is not connected to the negative terminal of...

1. The problem statement, all variables and given/known. Rank the following leaving group in order of increasing ability to leave? A) H2O B) NH2 C) OH D) I E) NH3 Homework Equations Also not entirely sure the order of H2O and NH3. What should I be looking for to answer this portion of the...

What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?

Is the metric tensor a tensor of rank two simply because the line element (or equivalent Pythagorean relation between differential distances) is "quadratic" in nature? This would be in opposition to say, the stress tensor being a tensor of rank two because it has to deal with "shear" forces. I...

Homework Statement In En the quantities Bij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components Cijkl and Dijkl for which ∑k ∑l Cijkl Bkl = Bij + Bji ∑k ∑l Dijkl Bkl = Bij - Bji are identities. Homework Equations The Attempt at a...

So that's the question in the text. I having some issues I think with actually just comprehending what the question is asking me for. The texts answer is: all 3x3 matrices. My answer and reasoning is: the basis of the subspace of all rank 1 matrices is made up of the basis elements...

Homework Statement Show that the matrix ##A## is of full rank if and only if ##ad-bc \neq 0## where $$A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$$ Homework EquationsThe Attempt at a Solution Suppose that the matrix ##A## is of full rank. That is, rank ##2##. Then by the rank-nullity...

Hi, Assume a matrix H n\times m, with random complex Gaussian coefficients with zero-mean and unit-variance. The covariance of this matrix (i.e., expectation [HHH]) assuming that m = 1 is lower than another H matrix when m > 1 ?? If this holds, can anyone provide a related reference? Thanks...

Hi I'm completely new in fortran programming and I just ran this code program array_test implicit none integer, parameter :: row = 3, column = 4 integer :: i, j integer, dimension(row: column) :: array2 open(1, file = "matrix.txt") array2 = reshape((/1,2,3,4,5,6,7,8,9,10,11,12/),(/row,column/))...

I read in many books the metric tensor is rank (0,2), its inverse is (2,0) and has some property such as ##g^{\mu\nu}g_{\nu\sigma}=\delta^\mu_\sigma## etc. My question is: what does ##g^\mu_\nu## mean?! This tensor really confuses me! At first, I simply thought that...

If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...

Hi guys. I'm studying an article on the measurements of entanglement in a pure bipartite state. I don't understand the definition of the Schmidt rank. It is equal to the rank of the reduced density matrix, isn't it? Is the Schmidt rank continuous and/or additive? I have no found on the net any...

Hey all 1. Two speakers emit sound in phase with the frequency of 13.2 kHz, the sound from the loudspeakers directed at points A , B and C. Rank the volume of the three points with the weakest first. speed of sound 340m/s 2. 3. I think I need to use Pythagoras says to calculate the path...

Hello! (Wave) A physical system is described by a law of the form $f(E,P,A)=0$, where $E,P,A$ represent, respectively, energy, pressure and surface area. Find an equivalent physical law that relates suitable dimensionless quantities. That' what I have tried so far: 1st step: Choice of...

Homework Statement Let Aijkl be a rank 4 square tensor with the following symmetries: A_{ijkl} = -A_{jikl}, \qquad A_{ijkl} = - A_{ijlk}, \qquad A_{ijkl} + A_{iklj} + A_{iljk} = 0, Prove that A_{ijkl} = A_{klij} Homework EquationsThe Attempt at a Solution From the first two properties...

In a given matrix A, the singular value decomposition (SVD), yields A=USV`. Now let's make dimension reduction of the matrix by keeping only one column vector from U, one singular value from S and one row vector from V`. Then do another SVD of the resulted rank reduced matrix Ar. Now, if Ar is...

Prove: Let $A$ be an $n$ by $n$ matrix of rank $r$. If $U={}\left\{X \in M_{nn}|AX=0\right\}$, show that $\dim U=n(n-r)$. Proof: Clearly, $\dim \operatorname{null}A=n-r$, and let $X=[c_1,c_2,...,c_n]$ where $c_i \in \Bbb{R}^n$ are column vectors. Then since $AX=0$, using block multiplication...

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of dimension $n-r$ in $\mathbf R^n$. We imitate the proof of Lemma 1 on pg 11 in Topology From A...

Homework Statement Matrix A is of size 5x3 (5 rows and 3 columns) with rank(A)=3. Find the reduced row echlon form of AThe Attempt at a Solution Rank(A)=3 thus, there are 3 pivot variables. Since there are 3 pivot variables and 3 columns=> no free variables, thus we have 2 rows of zeroes at...

Homework Statement Find Kernel, Image, Rank and Nullity of the matrix 1 −1 1 1  | 1 2 −1 1 | 0 3 -2 0 Homework EquationsThe Attempt at a Solution I have reduced the matrix into rref of 3 0 1 3 0 3-2 0...

Let S = [ x x x x ; a x x x ; 0 b x x ; 0 0 c x] Find the rank of S-K*identity. Attempt I basically did straight forward row reduction and got [ (x-k) x x x ; 0 ((x-k)^2-ax)/(x-k) x-ax/(1-k) a- ax/(1-k); 0 0 (x-k)-b[x(x-k)-ax]/ [(x-k)^2-ax]...

What I mean is can you list the progression of math classes or subjects that people can take in school based on how they are usually learned or what is more difficult, etc.? Like right now I'm taking high school Calculus and we had to do Pre-Calculus last year and Geometry and Algebra before...

Find rank and the bases for column and row spaces of the matrices 1 0 1 2 -1 3 3 -1 4 Now I can see instantly that row 3 is just row 1 + row 2 so it must be dependent. So that means that row 3 will turn to a row of zeros and thus the rank(A)=2 if I reduced matrix A to row echelon it...

Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...

It is pretty straight forward to prove that the Kronecker delta \delta_{ij} is an isotropic tensor, i.e. rotationally invariant. But how can I show that it is indeed the only isotropic second order tensor? I.e., such that for any isotropic second order tensor T_{ij} we can write T_{ij} =...

I'll be applying to PhD programs in pure math this fall. However, my profile isn't that outstanding, so I'll most likely being going to a "mid" to "low" rank (as determined by US News Grad Rankings, for what it's worth). Ultimately, I'd want to do research as well as teach lower/upper division...

Homework Statement Rank the electric potential energy at point ##P## for the following four cases: http://gyazo.com/c7d9df3d3d64cda909ddc0d2ab7686bc Homework Equations ##\Delta U_e = - W_∞## The Attempt at a Solution I believe it should be ##U_2 > U_1 > U_3 > U_4##, but I am not certain.

Homework Statement Consider a theory which is translation and rotation invariant. This implies the stress energy tensor arising from the symmetry is conserved and may be made symmetric. Define the (Schwinger) function by ##S_{\mu \nu \rho \sigma}(x) = \langle T_{\mu \nu}(x)T_{\rho...

The problem statement Let ##A ∈ K^{m×n}## and ##B ∈ K^{n×r}## Prove that min##\{rg(A),rg(B)\}≥rg(AB)≥rg(A)+rg(B)−n## My attempt at a solution (1) ##AB=(AB_1|...|AB_j|...|AB_r)## (##B_j## is the ##j-th## column of ##B##), I don't know if the following statement is correct: the columns of...

What does a rank of a matrix mean? why it is important? what does it tells you about?

Homework Statement Given that A is an 13 × 11 matrix of rank 3. a)The nullspace of A is a d-dimensional subspace of Rn. What is the dimension, d, and n of Rn? b) Is it true that for every b ∈ R11, the system ATx = b is consistent? (What does this means?) The Attempt at a...

Homework Statement Rank by acidity Homework Equations Br is lower on the periodic table, hence more acidic. But in the first compound, it is also farthest away from the carboxylic acid, which one gets priority ? The Attempt at a Solution I think The one with the bromo group closest to the...

Homework Statement I'm learning a bit about tensors on my own. I've been given a definition of a tensor as an object which transforms upon a change of coordinates in one of two ways (contravariantly or covariantly) with the usual partial derivatives of the new and old coordinates. (I...

There are two theorems from multivariable calculus that is very important for manifold theory. The first is the inverse function theorem and the second is the "constant rank theorem". The latter states that (Constant rank theorem). If ##f : U\subset \mathbb{R}^n \to \mathbb{R}^m## has...

Please see attached question In my opinion this question is conceptional and abstract.. For part a and b, I think dim(Ker(D)) = 1 and Rank(D) = n but I do not know how to explain them For part c What I can think of is if we differentiate f(x) by n+1 times then we will get 0 Can...

Tensors can be of type (n, m), denoting n covariant and m contravariant indicies. Rank is a concept that comes from matrix rank and is basically the number of "simple" terms it takes to write out a tensor. Sometimes, however, I recall seeing or hearing things like "the metric tensor is a rank 2...

I was wondering, is there a formula that converts a given z-score to its respective percentile rank? I know I can look up the conversion in a table, but I have a lot of data, and would rather just program a formula into Excel. Obviously, there is some sort of way that whoever created the table...

What is minimum possible rank of skew symmetric matrix ?

I'm studying low rank approximation by way of SVD and I'm having trouble understanding how the result matrix has lower rank. For instance, in the link the calculation performed on page 11 resulting in the so-called low rank approximation on page 12. This matrix on page 12 doesn't appear to me to...

Hello all I need to prove these 3 statements, and I don't know how to start... A is an nxn matrix: 1) if rank(A)=n then rank(adj(A))=n 2) if rank(A)=n-1 then rank(adj(A))=1 2) if rank(A)<n-1 then rank(adj(A))=0 thanks...:confused:

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...