# 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. ### 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. ### 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. ### 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...

16. ### 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. ### 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. ### 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. ### 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. ### 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. ### MHB Transpose Help Needed - Hi Everyone!

Hi Everyone, need some help to transpose (attached picts)Thanks very much in advance.
22. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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 BA, where BA=(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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...