What is Transpose: Definition and 96 Discussions

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations).The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.

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  1. N

    Are Eigen Vectors the Same for Eigen Values of Transpose A and A?

    A is a square matrix over F field if k is the eigen value of A prove that k is eigen value of A^t too and has the same eigen vectors ?? eigen vectors are the solution space P(A) is found by solving (A-kI)x=0 dim P(A)=dim n -dim (ro(a)) rho(a)=rho(a^t)...
  2. M

    Fortran Fortran Transpose: Transform a 4D Array into 2D

    Here is the problem.I have this array zmdsens(iper,i,1,iprd) where iper is period,i site,1 mt function and iprd conductivity.This array stores MT functions for all above mentioned.I need tot find transpose of MT function,but fortran 90 can easily do that only with 2dimensional arrays.How to...
  3. T

    Transpose of orthogonal matrix

    Homework Statement Orthogonal matrix means Q^{T}Q=I, but not necessary QQ^{T}=I, so why can we say the inverse of Q is Q^{T}? Homework Equations The Attempt at a Solution the attempt is actually in my question. It's something i don't understand when doing revision.
  4. B

    Linear algebra problem with transpose

    Homework Statement Find a formula for (ABx)T, where x is a vector and A and B are matrices of appropriate sizes. Homework Equations (AB)T = BTAT among a few others, probably the most relevant one with transposes here. The Attempt at a Solution I'm wondering what this "formula"...
  5. L

    Similarity transformation to the transpose

    I have a real nxn matrix A and I want to find P, so that P-1AP=AT. Does such a matrix exist? How do I find it? What if I have two matrices A,B. Does there exist a matrix P, that transforms both of them to their transposes? Thanks
  6. M

    What are the interpretations of the dual map in linear algebra?

    I'm sure there are a ton of ways to interpret what the transpose of a matrix represents. Could someone just give me a laundry list of interpretations? Thanks!
  7. matqkks

    What is the relationship between a matrix and its transpose in linear algebra?

    What is the geometric interpretation of the transpose of a matrix? Is there any physical significance of the matrix transpose?
  8. jinksys

    Linear Algebra: The transpose of A equals Inverse A, so

    If the transpose of A equals the Inverse of A, then det(A)=1. False. However, I don't follow the logic. If transA=InverseA, doesn't that mean the matrix is the identity matrix? The explanation says that det(A)= 1 and -1.
  9. B

    Calculating the Transpose of Adjoint of Dirac Spinor

    Homework Statement I want to compute the transpose of the adjoint of a Dirac spinor.Homework Equations My reasoning, based on learning Griffiths notation in “Intro to Elementary Particles”, p. 236, [7.58]: {\bar u^T} = {({u^\dag }{\gamma _0})^T} = {\gamma _0}^T{u^\dag }^T<mathop> =...
  10. D

    A transpose of a nonsingular matrix is nonsingular

    A transpose of a nonsingular matrix is nonsingular. This is true; however, how can this be done without using determinants? I know how to do this with determinants so please don't inform how to do this with determinants.
  11. S

    Proving eigenvalues = 1 or -1 when A = A transpose = A inverse A is circulant

    Homework Statement Prove all eigenvalues = 1 or -1 when A is circulant and satisfying A=A^T=A^-1 I can think of an example, the identity matrix, but i can't think of a general case or how to set up a general case. Homework Equations The Attempt at a Solution I can only show by...
  12. N

    Notation for Vector Transpose: \mathbf v

    Homework Statement Hi guys If I have a vector v, then is it correct notation to write \mathbf v = \left( {\begin{array}{*{20}c} {v_1 } \\ {v_2 } \\ \end{array}} \right) = (v_1,v_2)^T, where T is the transpose?
  13. Z

    Prove or disprove: A and A transpose have the same eigenspaces

    Homework Statement Prove or disprove: A and AT have the same eigenspaces. Homework Equations The Attempt at a Solution I know that A and AT have the same determinant and so they have the same characteristic polynomial and eigenvalues, but then if they are transposed then the...
  14. T

    Curl of the transpose of a gradient of a vector: demonstration of an identity

    I would like to demonstrate an identity with the INDICIAL NOTATION. I have attached my attempt. Please let me know where I made mistakes. Any suggestion? I am trying to understand tensors all by myself because they are the keys in continuum mechanics Thanks
  15. B

    Invertible Matrices and Rank 1 Matrices: Understanding Linear Transpose

    Homework Statement I have an idea on how to part 1, but I have no clue on how to do part 2 and 3. 1.Suppose A is invertible. Check that (A-1)TAT=I and AT(A-1)T=I, and deduce that AT is likewise invertible with inverse (A-1)T. 2. Suppose A is an mxn matrix with rank 1. Prove that there...
  16. Hepth

    Conjugate Transpose for a vector

    In particle physics, we commonly have the gamma matrices, whose conjugate transpose is the raised or lowered index. Does the same rule apply to ANY indexed quantity? What about to scalar/vectors like momentum. Is the conjugate of momentum: \left(q_\mu\right)^\dagger = q^\mu The...
  17. Z

    Transpose Problem: Solving Steering Geometry

    Hey guys, So I've actually learned a fair bit about trig identities the last few weeks and beginning to understand how they actually work thanks to Irrational, Mute and some prompting from Hurkyl. I'm still having trouble with transposing which I think should be fairly simple. The equation...
  18. Z

    What is the Transpose of Y = Sin(x) + Cos(x)?

    Hi Guys, Simple question; I'm trying to work out the transpose of Y = Sin(x) + Cos(x) to make x the subject. I thought it would be x = arccos(arcsin(y)) / 2 however I don't think that's right. Is there another theorem I'm missing?
  19. J

    Why the determinant of a matrix is equal to its transpose

    Homework Statement I don't understand why the determinant of a matrix is equal to its transpose...how is this possible? Homework Equations The Attempt at a Solution
  20. Saladsamurai

    MATLAB MATLAB save command and transpose command

    I have a bunch of row vectors saved as objects r1 r2 r3. I would like to send each set of data (row) to a text file, but I want it to save as a column. This means that I want to save the transpose of the data, i.e., r1' r2'... Unfortunately, when I try to use save r1.txt r1' -ascii MATLAB...
  21. C

    Minimal polynomial, transpose, similar

    Homework Statement a) Prove that if a polynomial f(lambda) has f(A)=0, then f(AT)=0 b) Prove that A and AT have the same minimal polynomial. c) If A has a cyclic vector, prove that AT is similar to A. 2. The attempt at a solution a) I know that I need to show that f(AT) =...
  22. D

    Fortran What is the most optimized way to transpose a matrix in Fortran?

    Hi! I'm working on a programming project(fortran 77). and I need to transpose a big matrix, and for the moment I'm doing it by to do-loops: DO 20 J = 2,NP DO 10 I = 1,J-1 T = P(I,J) P(I,J) = P(J,I) P(J,I) = T 10 CONTINUE 20...
  23. C

    Symmetric Matrix Eigenvector Proof

    Eigenvalue and eigenvector for a symmetric matrix Homework Statement Let A be a n by n real matrix with the property that the transpose of A equals A. Show that if Ax = lambda x, for some non-zero vector x in C(n) then lambda is real, and the real part of x is an eigenvector of A...
  24. A

    Proving the Properties of Pseudo Inverse and Transpose

    I have been battling with this for hours now, i just keep getting stuck. It is to show that: (xyT)+=(xTx)+(yTy)+yxT After expanding the left side, leting xyT=A. I get stuck at (yxTxyT)+yxT I have tried from both sides of the equation, but can't arrive at the expected result. Any clues?
  25. F

    Understanding Matrix Transpose and Examples | Learn about Matrix Transpose

    Hi Could somebody please tell me what the use is for the transpose of a matrix, and maybe give an example if possible. Thanks
  26. D

    What is the purpose of the transpose?

    Every book I've seen starts out with "to find the transpose, make B_ij = A_ji . However, they don't explain exactly why would would want to do this. Ie. they tell you the inverse is useful because if you have Ax = b, you can find x by writing b = A^{-1} x. The only thing I can think of to...
  27. I

    Transpose of Grassmanian variables

    Say, \psi^1,\ \psi^2 are Dirac spinors, and M is a matrix composed of Dirac matrices. Is the following equation hold? \bar{\psi^1}M\psi^2 = -\Big(\bar{\psi^1}M\psi^2\Big)^T I'm not quite sure, here is my derivation: \bar{\psi^1}M\psi^2 = \bar{\psi^1}_{i}M_{ij}\psi^2_j = -...
  28. M

    Help interpreting a differential transpose

    Hi, I'm currently trying to decipher an equation in a paper I'm reading for research of my own. However, I am running to a little trouble interpreting their notation and was hoping some of the knowledgeable people on this forum might be able to help. I have attached the image containing the...
  29. J

    What is the relationship between ker(A) and ker(A^TA)?

    B= A transpose What is the relation between ker(BA) and ker(A)? I was told that they are equal to each other, but I can't figure out why. ker(A) => Ax = 0 ker(BA) => BAx = 0 so that BA is a subset of A. This shows that ker(BA) =0 whenever ker(A) = 0, but how does this also show that...
  30. J

    Linear Algebra: Diagonalization, Transpose, and Disctinct Eigenvectors.

    Homework Statement Show that if an nxn matrix A has n linearly independent eigenvectors, then so does A^T The Attempt at a Solution Well, I understand the following: (1) A is diagonalizable. (2) A = PDP^-1, where P has columns of the independent eigenvectors (3) A is...
  31. D

    Vector/Matrix Differentiation: Where to Put the Transpose?

    if you want to find the derivative (gradient) of f(x)^2 when f is a vector, you would get 2*f(x)*del(f(x)) I never know where to put the transpose! sometimes its clear because another term in the equation will be a scalar, so you know an inner product is needed, but if you don't have a...
  32. H

    Q1: Does A and its transpose have the same eigenspace?

    So I've shown that A and A^T have the same char. polynomials => same eigenvalues, using the fact that detA = detA^T. I still can't see any way I could possibly show or disprove that the eigenspace is the same.
  33. T

    Proving transpose(AB)=transpose(B)*tranpose(A) Help Needed

    I need help on the tranpose of a multiple of a matrix. I need to prove: transpose(AB)=transpose(B)*tranpose(A) Any Ideas?
  34. M

    What is the missing piece in proving A=0 when A(A*)=0?

    Let A be a nxn matrix. Prove that if (A*)A=0 then A=0. What if A(A*) = 0? A* is the conjugate transpose of A. When I write out the expansion formula, I cannot conclude that every entry of A is zero. What am I missing?
  35. S

    MATLAB Matlab - finding the transpose of a matrix

    if i have a matrix A, and i want to find it's transpose on MATLAB i.e. i want to find A^T but how do i do that on matlab...what command do i type in?
  36. B

    Equivalence of Adjoint and Conjugate Transpose in Non-Orthogonal Bases?

    Is the adjoint of linear map only guaranteed to be equivalent to the conjugate transpose of the matrix when the matrix is taken with respect to an orthonormal basis? Is it sometimes still equivalent even when the basis is not orthonormal? For the problem I'm working on, I have...
  37. N

    Double dual/Double Transpose Question

    The question that I am stuck on is: Show that if X" (double dual of X) is identified with X and U" (double dual of U) with U via the duality relation, then T" (double transpose) = T. (Duality relation is f(L) = L (x) where f is in X", L is in X', and x is in X) So far, here is my work...
  38. S

    Prove that if A is nonsingular then the transpose of A is nonsingular.

    I haven't written a proof in 8 years. Linear Algebra proofs are going to be the death of me. I honestly don't know where to begin. I read a sort of primer on proof writing, but I could use a human walk through or some help. So far, I have: there exists a B such that AB = BA = I...
  39. A

    Symmetric Matrix Transpose: ABC^T ≠ CBA?

    (ABC)^T, A,B,C are all symmetric, then why isn't (ABC)^T = CBA? If you consider that (ABC)^T = (C^T)(B^T)(A^T) and in symmetrix cases, then C^T = C and so on...? (Latex edit by HallsofIvy)
  40. M

    Proof Help - Rank of the transpose of a Matrix

    Hi, I'm having trouble with a proof regarding the rank of the transpose of a matrix. Here's the question: Let A be an m x n matrix of rank r, which is of course less than or equal to min{m,n}. Prove that (A^t)A has the same rank as A. Where A^t = the transpose of A. I can easily...
  41. A

    Proving Matrix Transpose: (AB)^T = C^T = B^T * A^T

    how do you prove: (AB)^T=C^T=B^T*A^T?
  42. C

    Nullspace(kernel) and transpose

    hmmm...I have problems understanding this...how can the null space if a matrix(not necessarily a square) be the same as that of its transpose? Thanks in advance
  43. C

    Transpose of Matrix as Linear Map

    What are the relations between a matrix H and its transpose H^T? I am not asking about the relations between the coefficients, I am asking the relations as linear maps (H: F^m->F^n; H^T: F^n->F^m). I am not sure exactly how I should pose the question actually, but I am thinking there is some...
  44. F

    (LINALG) : Nullspace of transpose : N(A^T)

    I'm not sure if I am making a mistake, or my book is wrong, or if both answers are correct. But, it is confusing me, and I would like to know why. We are asked to find the basis of the following subspaces on the matrix A. Find: R(A^T),\,\,N(A),\,\,\,R(A),\,\,N(A^T) I'm having trouble...
  45. marcus

    Can we transpose the Cambridge Handbook into natural units?

    I am interested in a system of natural units that I see used more and more frequently in Quantum Gravity research papers so I like to try using them. the units are like conventional Planck except |8piG| = 1 I want to see if there is anything in the Cambridge Handbook of Physics Formulas...
  46. H

    Determinant and its transpose

    I've been doing revisions for my final exams, and I got stuck on the proof det A = det A^T, determinant of A = determinant of A transpose. How do I proof it?
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