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. H

    Proving ##(cof ~A)^t ~A = (det A)I##

    i-th column of ##cof~A## = $$ \begin{bmatrix} (-1)^{I+1} det~A_{1i} \\ (-1)^{I+2} det ~A_{2i}\\ \vdots \\ (-1)^{I+n} det ~A_{ni}\\ \end{bmatrix}$$ Therefore, the I-th row of ##(cof~A)^t## = ##\big[ (-1)^{I+1} det~A_{1i}, (-1)^{I+2} det ~A_{2i}, \cdots, (-1)^{I+n} det ~A_{ni} \big]## The I-th...
  2. M

    I Expressing the Matrix Transpose Function: Is There a Different Approach?

    One way to express a function of a matrix A is by a power series (a Taylor expansion). It is not too difficult to show that two functions f(A) and g(A) with such a power series representation must commute, i.e. f(A)g(A) = g(A)f(A). But matrices typically do not commute with their own transpose...
  3. Wrichik Basu

    Python Transpose of a non-square matrix (without using ndarray.transpose)

    While the prefix of the thread is Python, this could be easily generalised to any language. It is absolutely not the first time I am working with an array, but definitely the first time I am facing the task of defining the transpose of a non-square matrix. I have worked so much with arrays in...
  4. karush

    MHB Find $(AB)^T$: Calculate Matrix Product & Transpose

    Let $A=\left[\begin{array}{c}1 & 2 & -3 \\ 2 & 0 & -1 \end{array}\right] \textit { and } B=\left[\begin{array}{c}3&2 \\ 1 & -1 \\ 0 & 2 \end{array}\right]$ Find $(AB)^T$$AB=\left[ \begin{array}{cc}(1\cdot 3)+(2\cdot1)+(-3\cdot0) & (1\cdot2)+(2\cdot-1)+(-3\cdot2) \\ (2\cdot3)+(0\cdot1)+...
  5. Adgorn

    Proof regarding transpose mapping

    Homework Statement Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1. Homework Equations N/A The Attempt at a Solution Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So...
  6. Mr Davis 97

    Eigenvalues of transpose linear transformation

    Homework Statement If ##A## is an ##n \times n## matrix, show that the eigenvalues of ##T(A) = A^{t}## are ##\lambda = \pm 1## Homework EquationsThe Attempt at a Solution First I assume that a matrix ##M## is an eigenvector of ##T##. So ##T(M) = \lambda M## for some ##\lambda \in \mathbb{R}##...
  7. D

    I Transpose and Inverse of Lorentz Transform Matrix

    Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns. The defining relation (necessary and sufficient) for Lorentz transforms is...
  8. Mr Davis 97

    Show that A and its transpose have the same eigenvalues

    Homework Statement Show that ##A## and ##A^T## have the same eigenvalues. Homework EquationsThe Attempt at a Solution If they have the same eigenvalues, then ##Ax = \lambda x## iff ##A^T x = \lambda x##. In other words, we have to show that ##|A - \lambda I| = 0## iff ##|A^T - \lambda I| =...
  9. M

    I Transpose Property (where's my mistake)

    Hi PF! When proving ##\left(AB\right)^T = B^T A^T## I was thinking of writing ##\left(AB\right)_{ij} = A_{ik} B_{kj} = D_{ij}##. Then ##\left(D\right)^T_{ij} = D_{ji} = A_{jk} B_{ki} = A^TB^T## but clearly this is incorrect. Can someone tell me where my mistake is made? Thanks!
  10. tommyxu3

    I Proving Identity for Determinant of $A^tA$

    I have a problem of proving an identity about determinants. For ##A\in M_{m\times n}(\mathbb{R}),## a matrix with ##m## rows and ##n## columns, prove the following identity. $$|\det(A^tA)|=\sum_{1\le j_1\le ... \le j_n \le m} (det(A_{j_1...j_n}))^2$$ where ##A_{j_1...j_n}## is the matrix whose...
  11. D

    I What is the derivative of a matrix transpose?

    Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?
  12. Ismail Siddiqui

    [Linear Algebra] Conjugate Transpose of a Matrix and vectors in ℂ

    Homework Statement Let A be an n x n matrix, and let v, w ∈ ℂn. Prove that Av ⋅ w = v ⋅ A†w Homework Equations † = conjugate transpose ⋅ = dot product * = conjugate T = transpose (AB)-1 = B-1A-1 (AB)-1 = BTAT (AB)* = A*B* A† = (AT)* Definitions of Unitary and Hermitian Matrices Complex Mod...
  13. kostoglotov

    Transpose: a linear transformation?

    Alternate title: Is the textbook contradicting itself? imgur link: http://i.imgur.com/3sTVgwr.jpg But imgur link: http://i.imgur.com/33Ufncb.jpg So...it would appear that transposing has the property of linearity, but no matrix can achieve it...is transposing a linear transformation? The...
  14. PsychonautQQ

    Finding a matrix to represent a 2x2 transpose mapping

    Homework Statement Let L be a mapping such that L(A) = A^t, the transpose mapping. Find a matrix representing L with respect to the standard basis [1,1,1,1] Homework EquationsThe Attempt at a Solution So should I end up getting a 4x4 matrix here? I got 1,0,0,0 for the first column, 0,0,1,0 for...
  15. alyafey22

    MHB Confusion about the notation of transpose

    Define the following $$Z= \begin{pmatrix} 0 & A \\ B^T & T \end{pmatrix}$$ where we define $A$ and $B$ as $r \times m $ matrices and $T$ is an $m \times m$ matrix with nonzero distinct indeterminates at the diagonal, that is, $T_{i,i} = t_i$. What is the meaning of $B^T$ ?
  16. L

    Square matrix and its transpose satisfying an equation

    Homework Statement Show that if a square matrix A satisfies A3 + 4A2 -2A + 7I = 0 Mod note: It took me a little while to realize that the last term on the left is 7I, seven times the identity matrix. The italicized I character without serifs appeared to me to be the slash character /. then so...
  17. R

    I need to transpose for the value of Q

    B=PRn-Q(Rn-1)/R-1 Mod note: This thread is closed. @Rodo, this appears to be homework that is misplaced, with no effort shown. You are welcome to repost in the Homework & Coursework section, but you need to use the homework template and show what you have tried.
  18. S

    MHB What is the purpose of computing the transpose of a matrix?

    I am told to compute C^T .. what is this implying? I'm guessing maybe the transpose? Is this correct? Also should I post matrix related questions here or in the pre-calculus forum? This is a discrete mathematics class I am using these things in by the way.
  19. N

    Intuition & use of M*M^T product of matrix & its transpose?

    Hi all, I've occasionly seen people multiply a matrix by its transpose, what is the use and intuition of the product? Any help appreciated.
  20. P

    Transpose Inverse Property (Dual Vectors)

    Hello, While studying dual vectors in general relativity, it was written as we all know that dual vectors (under Lorentz Transformation) transform as follows: \tilde{u}_{a} = \Lambda^{b}_{a}μ_{b} where \Lambda^{b}_{a}= η_{ac}L^{c}_{d}η^{db} I was wondering if one can prove the latter...
  21. V

    MHB Transpose Help Needed - Hi Everyone!

    Hi Everyone, need some help to transpose (attached picts)Thanks very much in advance.
  22. N

    Nullspace of A transpose x: A Geometric Interpretation

    What does ATx=0 means? Does this notation means if A = [3,2;1,2;4,4], then, AT = [3,1,4;2,2,4] and ATx [x1;x2;x3] = 0? The nullspace of the transposed of the matrix A?
  23. N

    Find a basis for the null space of the transpose operator

    Homework Statement Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##. Homework Equations Let ##T:V\to W## be a linear...
  24. C

    Show that minimal poly for a sq matrix and its transpose is the same

    Homework Statement show that minimal poly for a sq matrix and its transpose is the sameHomework Equations The Attempt at a Solution no clue.
  25. Q

    Transpose of a Tensor Identity

    My textbook (regarding continuum mechanics) has the following identity that is supposed to be true for all tensors: a\cdotTb = b\cdotTTa But I don't get the same result for both sides when I work it out. For each side, I'm doing the dot product last. For example, I compute Tb first and...
  26. J

    The Transpose of a Matrix

    Let B=A^{T}A. Show that b_{ij}=a^{T}_{i}a_{j}. I have no idea how to approach this problem.
  27. E

    Transpose of the product of matrices problem

    Hi, The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get. I have, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix. I understand the equation...
  28. S

    Double transpose of a linear transformation

    I'm using a book that has a loot of errors (luckly most of them are easy to recognize, like a = instead of a ≠ or viceversa, but some are way more serious), and I'm not sure if it's a new error or a thing I don't understand. Either I didn't understood all the steps of the proof or the correct...
  29. M

    Regarding transpose of matrix products

    Starting out a Lin Alg class - my prof wrote this on the board. (ABC-1Dt)t = DC-1BtAt On the right hand side, I get why D is D, why A and B are now both transpose, but why is C still inverse? I know the rule (D-1)t = (Dt)-1, but I do not see how the heck it applies here or what would make the...
  30. Y

    MHB Proving Symmetry of Matrix Multiplication with Transpose | Step-by-Step Guide

    Hello I need to prove that for all matrices 'A', the multiplication of A with it's transpose, is a symmetric matrix. How should I do it ? Thanks !
  31. N

    Multiplying a vector with Square Matrix vs. its transpose

    Hi, I am new to Math so I am trying to get some intuition. Let's say I have a matrix A of n x n and a vector B of n x 1 what is the difference between A x B and A' x B? Thanks
  32. N

    Angle between vector and its transpose

    Hi What is the angle between a vector (e.g. a row vector A) and it's transpose (a column vector) ? I know what transpose means mathematically but what is the intuition? Thanks guys
  33. R

    Transpose a matrix whose elements are themselves matrices

    If I have (for simplicity) a vector ( A, B) where A and B are matrices how does the transpose of this look, is it ( AT, BT) or (AT BT)
  34. P

    Matrices: Transpose and Inverse

    Homework Statement Find (X * Y-1)T - (Y * X-1)T When X = [3 5] .....[1 2] and Y = [3 4] ...[2 3] Homework Equations Inverse= 1/ad-bc [d -b] ......[-c a] The Attempt at a Solution I got: [9 -6 ] [14 -9] But the answer is: [-3 -2] [6 3]I did the problem twice and got the same answer so I...
  35. V

    Square of transpose of two matrices

    Homework Statement Let A and B be two square matrices of order n such that AB = A and BA = B. Then, what is the value of [(A + B)t]2? Homework Equations The Attempt at a Solution [(A + B)t]2 = AtAt + AtBt + BtAt + BtBt. I tried to use the fact that AB = A and BA = B to keep...
  36. C

    Show that a matrix's transpose has same eigenvalue.

    Show that a matrix and its transpose have the same eigenvalues. I must show that det(A-λI)=det(A^t-λI) Since det(A)=det(A^t) →det(A-λI)=det((A-λI)^t)=det(A^t-λI^t)=det(A^t-λI) Thus, A and A^t have the same eigenvalues. Is the above enough to prove that a matrix and its transpose have the...
  37. B

    Quick matrix transpose proof help

    Homework Statement let transpose of A be noted by A` Show that if the matrix product AB is permitted, then so is the product B`A`, where B`A`=(AB)` Homework Equations C_{ij}=ƩA_{ik} B_{kj} where summing from k=1 to m A`_{ij} = A_{ji}The Attempt at a Solution It wants me to use the...
  38. L

    Dimension of the null space of A transpose

    So I'm given a matrix A that is already in RREF and I'm supposed to find the null space of its transpose. So I transpose it. Do I RREF the transpose of it? Because if I transpose a matrix that's already in RREF, it's no longer in RREF. But if I RREF the transpose, it gives me a matrix with 2...
  39. S

    Determinant of Transpose Operator

    I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's definition of the determinant expressed in terms of an alternating bilinear form but am having...
  40. G

    Matrix Addition and Transposition: How to Solve for Equal Variables

    Homework Statement here is the answer: The Attempt at a Solution I can't figure out how the matrix listed above in the answer is supposed to add up to -1. that's the only way that a and b can equal each other, that is, if they both add up to -1.
  41. G

    Multiplying a matrix by its transpose

    Homework Statement I don't see how you multiply a matrix by its transpose. If a matrix is 3 x 2 then its transpose is 2 x 3. I thought you couldn't multiply matrices unless they have the same rows and columns.
  42. M

    Dirac notation and conjugate transpose in Sakurai

    In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17): <B|X|A> = (<A|X^|B>)* = <A|X^|B>* where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X. My question is, since a bra is the conjugate...
  43. R

    Matrix Vector Transpose

    To find the least squares polynomial of degree 2 to approximate points (X,Y) given in the table X_____________Y 1_____________36 1.9_____________-49...
  44. E

    Transpose of a matrix with mixed indices

    Hi! Given a matrix A of elements A_i\;^j, which is the right transpose: A_j\;^i or A^j\;_i ?
  45. 3

    What Are the Eigenvalues of A Transpose A?

    Homework Statement Let A be an m x n matrix with rank(A) = m < n. As far as the eigenvalues of A^{T}A is concerned we can say that... Homework Equations The Attempt at a Solution If eigenvalues exist, then A^{T}Ax = λx where x ≠ 0. The only thing I think I can show is that...
  46. A

    Please can you help me transpose this problem?

    7.0588 = sin(10.4*∏*t) how do i transpose this to solve the equation finding a value for t? thanks
  47. W

    Proof that the transpose of a tensor is a tensor

    1Homework Statement Prove that the transpose of a tensor is a tensor. Homework Equations Definition of the transpose: a\bulletTb = b\bulletT^Ta where a and be are arbitrary vectors The Attempt at a Solution This isn't homework per se, I'm 60 yo and studing continuum mechanics...
  48. D

    Linear algebra matrices multiplication (transpose)

    Homework Statement We are looking for the matrix A Homework Equations (A^transpose)^transpose=A The Attempt at a Solution i would start with finding the transpose of the matrix. -5 0 -8 -7
  49. A

    Matrix similar to its transpose

    Why is every matrix (complex) similar to its transpose? I am looking at a typical jordan block and I see that the transpose of the nilpotent part is again nilpotent and actually similar to the nilpotent part. I can see that the scalar part of the jordan block does not change under...
  50. D

    Proving transpose of orthogonal matrix orthogonal

    Homework Statement Show that if A is orthogonal, then AT is orthogonal. Homework Equations AAT = I The Attempt at a Solution I would go about this by letting A be an orthogonal matrix with a, b, c, d, e, f, g, h, i , j as its entries (I don't know how to draw that here)...but...