rank Definition and 290 Threads

Homework Statement a)Let A and B be nxn matrices such that AB=0. Prove that rank A + rank B <=n. b)Prove that if A is a singular nxn matrix, then for every k satifying rank A<=k<=n there exists an nxn matrix B such that AB=0 and rank A + rank B = k. Homework Equations rank A + dim Nul...

How to prove that if A is a diagonalizable matrix, then the rank of A is the number of nonzero eigenvalues of A. Thanks and regard.

I'm trying to contract a rank 4 tensor with covariant rank 2 and contravariant rank 2 with four different indices [T[ab][cd]] to get a scalar value T and I have no idea how to do it as I'm sure a or b does not equal c or d. Any help would be much appreciated.

Homework Statement For any (nxn) matrix A, prove or disprove with a counter example: 1. Rank(A^2) <= rank(A) 2. Nullity(A^2) <= nullity(A) Homework Equations Rank = dimension of range Nullity = dimension of null space The Attempt at a Solution I have been trying a few examples...

Homework Statement Prove that similar matrices have the same rank. Homework Equations The Attempt at a Solution Similar matrices are related via: B = P-1AP, where B, A and P are nxn matrices.. since P is invertible, it rank(P) = n, and so since the main diagonal of P all > 0...

Homework Statement A is an c x d matrix. B is a d x k matrix. If rank(A) = d and AB = 0, show that B = 0.Homework Equations The Attempt at a Solution My textbook has a solution but I don't understand it: The rank of A is d, therefore A is not the zero matrix. (I asked my prof why d can't be...

I have a question about the rank of adjoint operator... Let T : V → W be a linear transformation where V and W are finite-dimensional inner product spaces with inner products <‧,‧> and <‧,‧>' respectively. A funtion T* : W → V is called an adjoint of T if <T(x),y>' = <x,T*(x)> for all x in V...

Hi, Does anyone know how to prove rank(A)=rank(AT A) where A is any matrix and AT is the transposed of matrix A? I have difficulty to prove the part that nulity(A)=nulity(AT A). Any help will be appreciated.

Hi Is it true that for an idempotent matrix A (satisfying A^2 = A), we have trace(A) = rank(A) Where can I find more general identities or rather, relationships between trace and rank? I did not encounter such things in my linear algebra course. I'm taking a course on regression...

Hey Guys, Another matrice question Homework Statement Prove: Rk(A+B)\leq Rk(A) +Rk(B) The Attempt at a Solution Rk(A+B) = Dim[R(A) + R(B)] Where R(A) is the row space of A we know that Dim[R(A)+R(B)] = Dim[R(A)] + Dim[R(B)] - Dim[R(A)\capR(B)] Which means that Dim[R(A)+R(B)]...

Homework Statement A sign is to be hung from the end of a thin pole, and the pole supported by a single cable. Your design firm brainstorms the six scenarios shown below. In scenarios A, B, and D, the cable is attached halfway between the midpoint and end of the pole. In C, the cable is...

Hey all. I know this is a basic concept but I don't really understand it. I don't get what the difference between rank and dimension is. According to my book, the rank of a matrix is the dimension of the column space. Does that not imply that they are the same, unless the question...

I was reading Turk and Pentland paper 'Eigenfaces for recognition' and they assert that, if M < N, the maximum rank of a covariance matrix is M - 1, being M the number of samples and NxN the size of the covariance matrix. Is there any simple demonstration of this fact? Thanks in advance...

Homework Statement Let A be an m * n matrix with rank m and B be an n * p matrix with rank n. Determine the rank of AB. Justify your answer. Homework Equations The Attempt at a Solution I don't really know where to start off, but I have some things that might help...

why not half-integer? according to the definition, such as [J_z,T^k_q]=q T^k_q it is quite possible that k can be a half-integer.

Homework Statement Find the rank of A = {[1 0 2 0] [4 0 3 0] [5 0 -1 0] [2 -3 1 1]} Homework Equations The Attempt at a Solution i row reduced A to be: {[1 0 0 0] [0 1 0 -1/3] [0 0 1 0]} where do i go from here?

I understand that there is a way to find a basis \{e_1,...,e_n\} of a vector space V such that a 2-vector A can be expressed as A = e_1\wedge e_2 + e_3\wedge e_4 + ...+e_{2r-1}\wedge e_{2r} where 2r is denoted as the rank of A. However the way that I know to prove this seems sort of...

Homework Statement Show that the rank 3 tensor S_{\alpha \beta \gamma}=F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta} is completely antisymmetric. I just don't feel comfortable doing this stuff at all. Each of the three terms seems like they should be...

HI, I came across the following question, which I could only solve for one trivial special case. I'm hoping for help from your side on how to deal with the general case. Assume we are in the situation that we have a decomposition of a full-rank d x d matrix, M, into a sum of N rank-1 matrices...

Hi everyone, (This isn't a homework problem). How does one show that there is no Lorentz invariant tensor of rank 3 and the only Lorentz invariant tensor of rank 4 is the 4D Levi Civita tensor? Thanks in advance.

Homework Statement The figure below shows cross sections through three long square conductors of the same length and material, with square cross sections of edge lengths as shown. Conductor B fits snugly within conductor A, and conductor C fits snugly within conductor B. Rank the following...

Homework Statement Prove that the three matrices have the same rank. \left[ \begin{array}{c} A\\ \end{array} \right] \left[ \begin{array}{c} A & A\\ \end{array} \right] \left[ \begin{array}{cc} A & A\\ A & A\\ \end{array} \right] Homework...

I'm using the MATMUL command to multiplicate two arrays: array A is of rank one and has three complex elements, while array B is a 3x3 matrix with complex elements. However, the compilation is aborted because "the shapes of the array expressions do not conform". I'm pretty sure that the...

Homework Statement Given rank(R) and a QR factorization A = QR, what is the rank(A) Homework Equations The Attempt at a Solution I want to know if multiplication by a full rank orthonormal matrix Q and an upper trapezoidal matrix R yields rank(R)=rank(Q*R)=rank(A) This is...

As a complete novice, I'm reading a text which says that a mixed second rank tensor T^{u}_{v} is reducible but don't see how. Anyone care to show me? :wink:

Homework Statement True or False: If A is an n x n matrix, then the rank of A equals the number of linearly independent row vectors in A. Homework Equations None The Attempt at a Solution Okay, I know this is a ridiculously easy question, but I'm wondering if there is a catch...

Homework Statement By considering the dimensions of the range or null space, determine the rank and the nullity of the following linear maps: a) D:Pn --> Pn-1, where D(x^k)=Kx^k-1 b) L:M(2,3) --> M(2,3) where L([a b c; d e f])=[d e f; 0 0 0] c) Tr:M(3,3) --> R, where Tr(A)=a11+a22+a33 (the...

Hi, My name is Jennifer and I'm new to Physics Forum. I was googling algebraic terms when I came across this site. It looks very helpful and I will greatly appreciate it if someone can help me answer this question:- Let L : Rn --> Rm and M : Rm --> Rp be linear mappings. a) Prore that rank(...

Homework Statement Find the rank and nullity of the given matrix: |-2 2 1 1 -2 |----->(1) |1 -1 -1 -3 3 |----->(2) |-1 1 -1 7 5 |----->(3) The attempt at a solution i know rank is the number of non-zero rows and nullity is the # of columns minus the rank matrix: i took...

What does it mean for a matrix to have rank 0 ( zero) ?

Theorem: Let A be an m x n matrix. If P and Q are invertible m x m and n x n matrices, respectively, then (a.) rank(AQ) = rank(A) (b.) rank(PA) = rank(A) (c.) rank(PAQ) = rank(A) Proof: R(L_A_Q) = R(L_AL_Q) = L_AL_Q(F^n) = L_A(L_Q(F^n)) = L_A(F^n) = R(L_A) since L_Q is onto...

Notations: L(V,W) stands for a linear transformation vector space form vector space V to W. rk(?) stands for the rank of "?". Question: Let τ,σ ∈L(V,W) , show that rk(τ + σ) ≤ rk(τ) + rk(σ). I want to know wether the way I'm thinking is right or not, or there's a better explanation...

Let T is a linear transformation from a vector space V to V itself. The dimension of V, denoted by dim(V), is finite. If the rank of T, denoted by rk(T), is equivalent to the rank of TT, i.e., rk(T)=rk(TT) why is the intersection of image of T(denoted by im(T)) and the kernel of T(denoted...

Homework Statement Determine the values of k, if any, that give the matrix (1,1,k),(1,k,1),(k,1,1) a rank of: zero, one, two, or three. Homework Equations The Attempt at a Solution I tried reducing to row echelon form but it's confusing dealing with all the k's. Is there a...

Can anybody show me how any isotropic rank 3 pseudotensor can be written as a_{ijk}=\lambda \epsilon_{ijk} for the isotropic rank 2 tensor case [i.e. a_{ij}=\lamda \delta_{ij} ], my notes prove it by considering an example i.e. a rotation by \frac{\pi}{2} radians about the z axis.

Here is the problem. Skip the first two paragraphs to get to the pure math part. There are two cubes of water, both with area of 2 x 2 dm^2. Via the bottom they are connected by a tube, and the flow of water is (surprise, surprise) proportional to the difference in water level between the...

what are the rank conditions for consistency of a linear algebraic system? my proffessor said that the coefficient matrix augmented with the column value matrix must have the same rank as the coefficient matrix for consistency of the system of equations. however does the term rank apply to...

Given a Riemann Curvature Tensor. How do you know the weight and rank of each: R^{i}_{jki} R^{i}_{jik} R^{i}_{ijk} Is the Ricci tensor always a zero tensor for diagonal metric tensors?

Let R be a commutative ring with 1. If F is a free module of rank n < 1, then show that HomR(F;M) is isomorphic to M^n, for each R-module M. I was thinking about defining a map Psi : HomR(F;M)--> M^n by psi(f) = (f(e1); f(e2); ... ; f(en)) where F is free on (e1; ... ; en) and show Psi is...

My question is let the linear mapping T : R2->R3 be given by T(x,y)=(x-y,2y-2x,0) write down bases for its image and null-space and determine its rank and nullity. Find the matrix A that represents T with respect to the standard bases of R2 and R3 now i think i know how to do this but I'm...

Hi everyone, I've a simple question (the answer may be so trivial that I really ought to be ashamed for asking!) in elementary matrix theory: "Does there exists any relation between the number of non-zero eigen values of a matrix with its rank?" The matrix is taken to be a general (square, of...

Please could someone help me with this question, thank you. Find dim[Ker(D^2 -D: P_3(F_3) ==>P_3(F_3))] Where dim is dimension, Ker is kernal D is the matrix 0100 0020 0003 0000 D^2 is the derivative of D is it equals 0020 0006 0000 0000 And F_3 is the field subscript3...

When one want to find selection rules for matrix element of (for example) electric quadrupole moment tensor Qij, irreducible components of Qij are needed to apply Wigner-Eckart theorem. When symmetry group is SO(3) irreducible component can be found using what we know from addition of angular...

I'm trying to show that any matrix X with rank n can be written as the sum of matrices Z and Y with rank n-1 and 1, respectively. Since X,Y, Z have the same dimensions, is this a simple matter of saying pick one of the columns in X with a pivot. Let Z= X with this column replaced by zeroes...

How can you prove that matrix X with rank n can be written as the sum of matrices Y and Z where Y has rank n-1 and Z has rank of 1. Thanks!

If A is an nxk matrix of real numbers (n>=k) of rank k, is it true that we can eliminate n-k lines of A to obtain a matrix A' of nonvanishing determinant? I convinced myself of that one time while in the bus and now I can't find the proof.

I've been tasked with proving the existence of a full rank factorization for an arbitrary m x n matrix, namely: Let \textit{A} \in \textbf{R}^{m x n} with \textit{rank(A) = r} then there exist matrices \textit{B} \in \textbf{R}^{m x r} and \textit{C} \in \textbf{R}^{r x n} such that \textit{A =...

hello again, I'm once again stumped, i was asked to find the rank and nullity of the projection u onto v so here is the given: T(u)=ProjvU, where v = <2,4> and this is what i did: let u = <u1 , u2> and plugged everything in the projection formula and ended up with < 4 + 2(u1) , -16 +...

[SOLVED] Proving col rank = row rank Homework Statement Demonstrate these four assertions to get an alternate proof that column rank equals row rank. (a) \vec{y}\cdot\vec{y} = \vec{0} \Leftrightarrow \vec{y} = \vec{0} (b) A\vec{x} = \vec{0} \Leftrightarrow A^TA\vec{x} = \vec{0} (c) \dim...

can anyone help me with this proof rank of two similar matrices is same.