What is rank: Definition and 307 Discussions

Receptor activator of nuclear factor κ B (RANK), also known as TRANCE receptor or TNFRSF11A, is a member of the tumor necrosis factor receptor (TNFR) molecular sub-family. RANK is the receptor for RANK-Ligand (RANKL) and part of the RANK/RANKL/OPG signaling pathway that regulates osteoclast differentiation and activation. It is associated with bone remodeling and repair, immune cell function, lymph node development, thermal regulation, and mammary gland development. Osteoprotegerin (OPG) is a decoy receptor for RANKL, and regulates the stimulation of the RANK signaling pathway by competing for RANKL. The cytoplasmic domain of RANK binds TRAFs 1, 2, 3, 5, and 6 which transmit signals to downstream targets such as NF-κB and JNK.
RANK is constitutively expressed in skeletal muscle, thymus, liver, colon, small intestine, adrenal gland, osteoclast, mammary gland epithelial cells, prostate, vascular cell, and pancreas. Most commonly, activation of NF-κB is mediated by RANKL, but over-expression of RANK alone is sufficient to activate the NF-κB pathway.RANKL (receptor activator for nuclear factor κ B ligand) is found on the surface of stromal cells, osteoblasts, and T cells. Mutations affecting RANK have been associated with infantile malignant osteopetrosis in humans, mice and cats.

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  1. T

    Understanding Low Rank Approximation with SVD: A Comprehensive Guide

    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...
  2. Y

    MHB Prove These 3 Statements About A Matrix: Rank of Adjoint

    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:
  3. G

    MHB QR Decomposition and Full Column Rank 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...
  4. M

    Rank the material according to their indices of refraction

    Homework Statement In the figure below (see image in color), light travels from material 'a', through three layers of other materials with surfaces parallel to one another, and then back into another layer of material 'a'. The refractions (but not the associated reflections) at the surfaces...
  5. Petrus

    MHB 4x4 Matrix with rank B=4 and B^2=3

    Hello MHB, "Can we construct a 4x4 Matrix B so that rank B=4 but rank B^2=3" My thought: we got one condition for this to work is that det B=0 and det B^2 \neq 0 and B also have to be a upper/lower or identity Matrix. And this Will not work.. I am wrong or can I explain this in a better way...
  6. Petrus

    MHB Can You Find a Matrix with Ranking Requirements?

    Hello MHB, I would like to have a tips for this problem. Find a matrice A of the order 4 x 4 satisfying that rank A=3, rank A^2=2, rank A^3=1 and rank A^4=0 I have no idé how I should think and to try guess the matrice don't fel correct.. |\pi\rangle
  7. B

    Does the characteristic polynomial encode the rank?

    Similar matrices share certain properties, such as the determinant, trace, eigenvalues, and characteristic polynomial. In fact, all of these properties can be determined from the character polynomial alone. However, similar matrices also share the same rank. I was wondering if the rank is...
  8. F

    Admissions Influence of undergraduate class rank on gradute admission

    I'm currently a Computer Engineering undergraduate student at the State University of Campinas (Brazil), and according to Times Higher Education World University Rankings, my university is ranked second best in Latin America. I'm considering to pursue a graduate level education on a top-tier...
  9. B

    Operation to make an (m+n)th rank tensor of rank-m and rank-n tensors

    Homework Statement We know that c[ij] = a[i]b[j] is a way to make a rank-2 tensor from two rank-1 tensors. We also know that C[abcxyz]=A[abc]B[xyz] is a way to make a rank-6 tensor from two rank-3 tensors. However, is there a matrix representation of this? I know the idea of a 6-dimensional...
  10. D

    How to count Spearman Rank order correlation

    Homework Statement calculate the rank order correlation between the following data: 6, 5, 4, 2, 3, 3, 8, 3, 7, 6, 7, 5, 5, 4, 2, 7, 6, 2, 4, 6 4, 3, 6, 7, 6, 7, 1, 9, 1, 2, 3, 4, 5, 5, 7, 1, 2, 9, 5, 4 Homework Equations Following the output from...
  11. C

    Block triangular matix has rank >= ranks of diagonal blocks?

    Hey! I found this interesting theorem in a textbook, but I was unable to find a proof for it neither in the web nor on my own Homework Statement The rank of a block triangular matrix is at least and can be greater than the triangular blocks. proof? specificaly, look here: pp. 25...
  12. I

    Form of symmetric matrix of rank one

    Homework Statement The question is: Let C be a symmetric matrix of rank one. Prove that C must have the form C=aww^T, where a is a scalar and w is a vector of norm one. Homework Equations n/a The Attempt at a Solution I think we can easily prove that if C has the form...
  13. I

    MHB Form of symmetric matrix of rank one

    The question is:Let $C$ be a symmetric matrix of rank one. Prove that $C$ must have the form $C=aww^T$, where $a$ is a scalar and $w$ is a vector of norm one.(I think we can easily prove that if $C$ has the form $C=aww^T$, then $C$ is symmetric and of rank one. But what about the opposite...
  14. B

    Eigenvalues of a rank 1 matrix?

    How come a square matrix has eigenvalues of 0 and the trace of the matrix? Is there any other proof other than just solving det(A-λI)=0?
  15. M

    Fortran Rank mismatch in argument (Fortran 90)

    Hello everyone, i am dealing with the code which can help me to solve fluid dynamics problems with using LBM methods. Anyways, since i am beginner on Fortran i couldn't solve the rank mismatch error, i think it is easy one but i just can't fix it, i am waiting for your help. Here is the problem...
  16. B

    Fortran Fortran bug: rank problem gfortran

    This is my own code, and it won't compile with gfortran. All I want to do is extract the location of the cell with the minimum value in an array. A seemingly simple task but one that does not work with the intrinsic function minloc, for reasons I do not understand. The error message...
  17. S

    I don't understand why the rank = n - Rank-nullity theorem - nullity

    I don't understand why the rank = n -- Rank-nullity theorem -- nullity Homework Statement I'm working on #1 (the solutions are also included in that pdf) here ( http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/exam-1/MIT18_06SCF11_ex1s.pdf )...
  18. S

    MHB Problem of Rank of a matrix

    If rank of A is 2. Is it possible to find the rank of A+A2+A3+A4 from that information? Please help
  19. G

    The rank of a block matrix as a function of the rank of its submatrice

    Hello everyone, I would like to post this problem here in this forum. Having the following block matrix: \begin{equation} M=\begin{bmatrix} S_1 &C\\ C^T &S_2\\ \end{bmatrix} \end{equation} I would like to find the function $f$ that holds rank(M)=f( rank(S1), rank(S2)). S_1 and S_2 are...
  20. S

    How to calculate rank of 2 by 1 matrix?

    how to calculate rank of 2 by 1 matrix..?? hey guys so i am well familiar with finding out rank of square matrices but if matrix is just a row or column vector then how to determine its rank..considering the example below: a=[x1 x2 x3] where is column matrix while x1,x2,x3 are...
  21. S

    Rank of a matrix and max number of missing values

    Hello all, I have a question: assume in matrix M(n*n), each element M(i,j) of matrix is computed as M(i&)*M(&j) / M(&&) where M(i&) is the summation of ith row, and M(&j) is the summation of jth column and M(&&) is the summation of all M(ij) for i=1..n and j=1..n. Now I want to know what is...
  22. S

    Schools Does University Ranking Impact Post-Graduate Studies?

    I am choosing between a rank 24 uni and a rank 45 uni, should rank have any effect on my decision?
  23. E

    Rank of Matrices and Eigen Vectors

    Homework Statement Find the rank off matrices? i)A=[2 0 9 2; 1 4 6 0; 3 5 7 1 ] 3X4 ii)A=[3 1 4; 0 5 8; -3 4 4; 1 2 4;] 4X3 Find Eigen Vectors and Values of A; A = [3 2 4; 2 0 2; 4 2 3 ] Homework Equations -when det(A) is not equal to zero it will the rank of matrices...
  24. A

    Inner product of rank 2 tensor and a vector

    I been reading some material that lead me to understand that it takes an inner product of a dyad and a vector to obtain another vector at an angle to the initial one... cross product among two vectors would be an option only if we are willing to settle to a right angle. After few days i...
  25. E

    Use SVD to show rank(XGY) = rank (G)

    1. Use the Singular Value Decomposition (SVD) of G to prove: rank(XGY^T) = rank (G) Given that X and Y are two full column-rank matrices, but may not have the same rank. 2. The attempt at a solution \begin{eqnarray*} XGY^T & = & X(U\Sigma V^T)Y^T \\ & = & XU \left(...
  26. S

    MHB Rank & Nullity: 3x3 Matrix w/ Plane Origin & LD Vectors

    1.(a) Give an example of 3*3 matrix whose column space is a plane through the origin in 3-space (b) what kind of geometry object is the null space and row space of your matrix 2. Prove that if a matrix A is not square, then either the row vectors or the column vectors of A are linearly dependent.
  27. T

    Rank of a Matrix and whether the columns span R12

    Homework Statement Let M be the 12 x 7 coefficient matrix of a homogeneous linear system, and suppose that this system has the unique solution 0 = (0, ..., 0) \in ℝ7. 1. What is the rank of M. 2. Do the columns of M, considered as vectors in ℝ12, span ℝ12. Homework Equations The...
  28. djh101

    Courses How would you rank these math courses in terms of difficulty?

    I'm trying to keep a balance in difficulty next quarter and would appreciate some feedback as to the difficulty of these classes. The four I'm choosing between are Linear Algebra (upper division), Linear and Nonlinear Systems of Differential Equations, Ordinary Differential Equations, and...
  29. S

    Finding rank and nullity of a linear map.

    Homework Statement let a be the vector [2,3,1] in R3 and let T:R3-->R3 be the map given by T(x) =(ax)a State with reasons, the rank and nullity of THomework Equations The Attempt at a Solution Im having trouble understanding this... I know how to do this with a matrix ie row reduce and no. of...
  30. J

    MHB Rank & Letter of $\bf{\mathbb{INDONESIA}}$ in Dictionary

    If all the letters of the world $\bf{\mathbb{INDONESIA}}$ are arrange in a English Dictonary, Then $(a)\;\; $ Rank of The word $\bf{\mathbb{INDONESIA}}$ $(b)\;\; 61^{th}$ Letter in Dictonary araanging $\bf{\mathbb{INDONESIA}}$
  31. O

    Rank the velocities of the balls

    Homework Statement Small masses m1 (m1 = 30 kg) and m2 (m2A = 5 kg; m2B = 10 kg; m2C = 40 kg; m2D = 50 kg; m2E = 30 kg) are each attached to a string of length 2.0 m. The other end of each string is attached to a common point on the ceiling. The masses are raised until each string is at an...
  32. V

    Rank Velocity Vectors by Kinetic Energy

    Rank the following velocities according to the kinetic energy a particle will have with each velocity, greatest first: (a) v = 4i +3j, (b) v = -4i +3j, (c) v = -3i + 4j, (d) v = 3i - 4j, (e) v = 5i, and (f) v = 5 m/s at 30 degrees to the horizontal. K = 1/2mv^2 I am not sure how to...
  33. N

    Proof involving Rank Nullity Theorem

    I hope I'm posting this in the right place. Homework Statement Let V be a finite dimensional vector space over a field F and T an operator on V. Prove that Range(T^{2}) = Range(T) if and only if Ker(T^{2}) = Ker(T) Homework Equations Rank and Nullity theorem: dim(V) = rank(T) +...
  34. G

    How to rank random function from smallest to largest with inverse f included?

    Homework Statement The graph of y=f(x) is shown below. http://Newton.science.sfu.ca/cgi-bin/plot.png?file=public_public_1346904771_18810161_plot.data Rank the following from smallest(1) to largest(4). f−1(0) f(0) f(5) f−1(5) Homework Equations none available The...
  35. J

    What is the rank of an nxn matrix and how is it determined?

    Homework Statement http://img94.imageshack.us/img94/5227/nxnmatrix.png Homework Equations Rank(A) = the number of pivots in Matrix A. The Attempt at a Solution I've spent some time rewriting the matrix and other operations. I really just feel like I'm banging my head against the wall. Not...
  36. Q

    What is the Rank of the Adjugate Matrix?

    how that the rank of the adjugate matrix (r(adj(A))) is : n if r(A)=n 1 if r(A)=n-1 0 if r(A)<n-1 How to deal with the proof? Can someone give more insight? What proof should I use here? I have an idea only for the third statement.
  37. B

    FEM: Rank deficiency and hourglassing

    Hello, I am having somewhat difficulty understanding the concepts of rank deficiency and hourglassing in finite element methods. Essentially, I have been reading a book outlining this very briefly on half a page and I need a bit more information. As an example: If we have a 2D elasticity...
  38. matqkks

    Are there any real life applications of the rank of a matrix? It need

    Are there any real life applications of the rank of a matrix? It need to have a real impact which motivates students why they should learn about rank.
  39. matqkks

    MHB Matrix Rank: Real-Life Applications & Motivation

    Are there any real life applications of the rank of a matrix? It need to have a real impact which motivates students why they should learn about rank.
  40. Saitama

    Rank Order of SN2 Reactivity: CH3-Cl vs CH3-CO-CH2-Cl

    Homework Statement The decreasing order of rate of SN2 reaction is: a)CH3-Cl b)CH3-CO-CH2-Cl Homework Equations The Attempt at a Solution I have been trying hard to find the reason why i am wrong. It's obvious that less hindrance, more reactivity towards SN2. Using the same...
  41. caffeinemachine

    MHB Linear algebra. Rank. linear independence.

    Let $V$ be a finite dimensional vector space. Let $T$ be a linear transformation on $V$ with eigenvalue $0$. A vector $v \in V$ is said to have rank $r > 0$ w.r.t eigenvalue $0$ if $T^rv=0$ but $T^{r-1}v\neq 0$. Let $x,y \in V$ be linearly independent and have ranks $r_1$ and $r_2$ w.r.t...
  42. A

    MHB Rank of the product of two matrices

    Hello Both of the below theorems are listed as properties 6 and 7 on the wikipedia page for the rank of a matrix. I want to prove the following, If A is an M by n matrix and B is a square matrix of rank n, then rank(AB) = rank(A). Apparently this is a corollary to the theorem If A...
  43. M

    School rank or school scenery?

    I am at community college right now doing my TAG (transfer admission guarantee) to a UC school. The problem is you only get 1 school to TAG to. I have been thinking about UC Davis, mostly because I've lived in southern california all my life and wanted a change of scenery. The other schools I...
  44. S

    A compact, bounded, closed-range operator on a Hilbert space has finite rank

    Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...
  45. A

    Understanding Rank One Matrices and Their Application in Nullspace and Row Space

    Hi: I see an principle about rank one matrice in the book, and it says if u=(1,2,3), \nut=[1 3 10], with Ax=0, the equation \nutx=0; The problem is I see an example like following: s1=[-3 1 0] s2=[-10 0 1] The nullspace contains all combination of s1...
  46. F

    Relationship between eigenvalues and matrix rank

    I'm looking into the stability of a system of ODEs, for which we've mannaged to extract a Jacobian matrix. Two of our eigenvalues are within our nummerical error tolerance, but they are close to zero. One of them is positive, which poses a problem for our stability analysis. We do know that...
  47. H

    Rank of the commutator.

    I found this theorem on Prasolov's Problems and Theorems in Linear Algebra: Let V be a \mathbb{C}-vector space and A,B \in \mathcal{L}(V) such that rank([A,B])\leq 1. Then A and B has a common eigenvector. He gives this proof: The proof will be carried out by induction on n=dim(V). He...
  48. H

    Rank, Co-factor matrices.

    Homework Statement Let A be an n x n matrix where n \geq 2. Show that A^{\alpha} = 0 (where A^{\alpha} is the cofactor matrix and 0 here denotes the zero matrix, whose entries are the number 0) if and only if rankA \leq n-2 Homework Equations The Attempt at a Solution No idea...
  49. P

    Power method to rank baseball teams

    Use the power method to rank the baseball league with the matrix { 1, .5, .5} {.5, 1, 1/3} {.5, 2/3, 1} So I choose some random matrix which sum to one so let x={.5,.3,.2}^T So { 1, .5, .5} {.5, 1, 1/3} {.5,.3,.2}^T= X_{1} {.5, 2/3, 1} And I keep repeating this...
  50. G

    Problem with Elementary row operations and rank theorems.

    Ok, so I am taking my first course in linear algebra, and even though I am not a math major (physics major actually), I can't help but wish my teacher and text were more rigorous. So let me start by telling you all the problem I am having: (First question) My book states the following...
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