Questions tagged [transpose]

In linear algebra, the transpose of a matrix is another matrix whose i-th row and j-th column is the j-th row and i-th column of the original matrix.

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Orthogonally equivalent matrices

Suppose $A$ is any square matrix, then how do I show that there exists two orthogonal matrices $Q$ and $R$ such that $Q^TAR=A^T$? I can show this when $A$ has distinct eigenvalues, but how do I show ...
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Covariance matrix multiplied by its transpose

Suppose we have a covariance matrix $A$ (it is automatically symmetric). So, $A^TA$ is also symmetric. However, do the elements of the matrix $A^TA$ keep the same pair-values as that of the matrix $A$...
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If $X \mapsto X^tG$ is an isomorphism, then $G$ is an invertible matrix. (Bilinear forms)

Let $f : E \times F \to R$ be a bilinear form that is non-singular on the left, where $E, F$ are free $R$-modules of dimension $n$ (both of them). Then if $X, Y$ are column vectors for general ...
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How do I find A if I have the transpose of A?

So I'm thinking about this problem I know that $(A^T)^T=A$ but when I have a coefficient like in this problem above, what do I do?
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What does it mean to have the transpose symbol on the solution set of a linear system?

I know that the superscript $T$ on a matrix indicates that the matrix is to be "transposed", that is its columns turned into rows; however for some reason I am seeing it on some solution ...
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Determinant equals 1 whereas Inverse unequal to the transpose?

So briefly: Given the matrix $A$ with det$(A) = 1$: $\left(\begin{array}{ccc} 1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1 \end{array}\right)$ Its inverse however is set by $\left(\begin{...
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What is the difference between a vector and the transpose of the vector [duplicate]

I would like to understand what the transpose of a vector represents different from the vector itself. I am not looking for a proof, just an explanation with some examples. I understand how the ...
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A matrix equation $Ax=0$ has infinite solutions, does $A^Tx = 0$ have infinite solutions?

I'm wondering whether a system with a transpose of a matrix has the same type of solution that the original matrix system has. If an equation $Ax=0$ equation has a unique solution, would a system with ...
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What are the requirements on a matrix $A$ for $A^TA$ to have an inverse?

I know there's a theorem stating that if the columns of $A$ are independent, then the columns of $A^TA$ are also independent ($A$ is invertible $\Rightarrow A^TA$ is positive symmetric definite is the ...
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Proof that $AA^T$ is invertible if $A$ has independent rows

While looking for the answer for my question I came across this post. It may be a silly idea, but if $A^{t}$ has independent rows can I just transpose it and get $A$ with independent columns and ...
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Conclude that $A^TA=I$, so $A^{-1}=A^T$.

A matrix $A$ is said to be orthogonal if for every vector $\textbf{x}$ we have $||A\textbf{x}||_2=||\textbf{x}||_2$. Show that such a matrix will necessarily have the property that $\left<A\...
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Transpose of a 3D Tensor [duplicate]

I'm finding it difficult to wrap my head around how the transpose operation works for Tensors of Rank 3 and above. Here's an example in PyTorch I was doing a transpose of tensors of rank 3 and ...
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Can $RanA=KerA^T$ for a real matrix $A$? And for complex $A$?

My question is divided in two: $\DeclareMathOperator{\Ran}{Ran}\DeclareMathOperator{\Ker}{Ker}\DeclareMathOperator{\span}{span}$ $1$- Can $\Ran A=\Ker A^T$ for real matrices? $2$ - What happens with ...
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Rewriting M in terms of omega?

Given a vector $v \in \mathbb{R}^n$ then show that we can express $\sum_{k} \omega_kv^2_k$ as a matrix product of the form $v^TMv$. Give an expression for $M$ in terms of $\omega = [\omega_1 ... \...
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Question regarding commutativity of matrices

Given, $A, B, C$ be $2 \times2$ matrices with entries from the set of real numbers. Define $A*B = \frac{1}{2}(AB'+A'B)$ where $X'$ denotes the transpose of $X$. Does $A*B=B*A$? My book answers this ...
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Are the entries of diag($xx^T$) given by $x^Tx$?

If $x$ is a $n \times 1$ vector, are the entries of diag($xx^T$) given by $x^Tx$?
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Determinant, same for transpose?

$$\det(A) = \det(A^\top)$$ Wiki confirms this. How can I see this this is true? I tested out a few examples and it seems to hold, however unsure the exact reasoning. For instance: $$\begin{vmatrix} ...
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Proving $(AB)^T=B^TA^T$: Can you find the flaw in the reasoning?

Proving $(AB)^T=B^TA^T$ isn't difficult. An example of such a proof can be found here: http://www.math.ucdenver.edu/~esulliva/LinearAlgebra/ABT.pdf However, if I take a slightly different approach, it ...
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Is the matrix used to prove independence of $\bar Y $ and $S^2$ for a random normal sample orthogonal?

I'm looking at a proof that $\bar Y $ and $S^2$ for a sample of $n$ random normal variables are independent using two different sources. Both sources makes use of identical matrices linked in the ...
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Closed Image and finite cokernel $\Leftrightarrow$ Closed image of the transpose and finite dimensional kernel

Let $D:X\rightarrow Y$ be a bound linear operator. $\newcommand{\im}{\operatorname{im}}$ The claim given is: A closed image $D(X)$ and finite dimensional cokernel is equivalent to a closed image $D^*(...
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Can you use matrices to solve for gravitational potential energies

I was wondering whether its possible to use matrices to calculate the gravitational potential energy of a system. I know it's possible to calculate the acceleration due to gravity on a system by ...
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Show that if we have a matrix $A$, which can be written as $A=DD^T$ for some matrix $D$, then…

we have $\bar{x}^TA\bar{x}\geq 0$ for any column vector $\bar{x}$. I do not really know where to start, but I do know that I can somehow show the scalar $\bar{x}^TA\bar{x}$ is equal to $||D^T\bar{x}||^...
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Find number of skew-symmetric matrices of order $3\times 3$ in which all non-diagonal elements are different

Find number of skew-symmetric matrices of order $3\times 3$ in which all non-diagonal elements are different and belong to the set $\{-9,-8,-7,...,7,8,9\}$ My Attempt: I did a simple calculation and ...
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Property of product of matrices with their transpose

Let $A$ and $B$ be matrices of size $m\times n$. Claim: $A^TA=B^TB$ iff there exists an orthogonal matrix $Q$ such that $QA=B$. The backward implication is trivial, consider $B^TB=(QA)^T(QA)=A^TQ^TQA=...
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Prove that every orthogonal matrix has a null space {0}

I am currently working on some practice problems regarding orthogonality and its properties, and one of the proofs I am trying requires that I show prove that "every orthogonal matrix has a null ...
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What is the condition for “transpose(A) x B = transpose(B) x A” to be true?

I need help with some basic doubts in linear algebra. Given two vectors of equal dimensions, A and B, it can be easily shown that "transpose(A) x B = transpose(B) x A". What if A and B are ...
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How to show that $A′A$ is a positive definite matrix given rank$(A) = k$?

I have difficulty in trying to prove the following question.Let $A = n \times k, k ≤ n$. Show that, if rank$(A) = k$, then $A′A$ is a positive definite matrix ($A'$ denotes the transpose of $A$). Do ...
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Does the expression $\frac{\mathbf{v}\mathbf{v}^T}{\mathbf{v}^T\mathbf{v}}$ have a name?

I have encountered the matrix $$\frac{\mathbf{v}\mathbf{v}^T}{\mathbf{v}^T\mathbf{v}}$$ (where $\mathbf{v}$ is a column-vector) a several times when differentiating vector expressions and was ...
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How to modify a formula to suit this application

I have an problem that I am unsure how to solve. Hopefully someone can edit this question to get the fancy equations up as I don't know how. I have a formula to find a variable, say $$a = \frac{(b * c)...
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Behaviour of orthogonal matrices

I am given that A is an orthogonal matrix of order $n$, and $u, v$ are Vectors in the $R^n $ space. I need to prove that $||u|| = ||Au||$. The first step of the solution hint I am given is that $$||Au|...
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SVD: Matrix size mismatch: Part 1

I'm reading this article. In the "Proof" section, there is an equation: $A^T = (USV^T)^T = VS^TU^T = VSU^T$ I don't understand the $VS^TU^T = VSU^T$ part. Is this $VSU^T$ simply a typo? If ...
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Inverse matrix of a special matrix

$\vec{u}$, $\vec{v}$ and $\vec{w}$ form a matrix $M$. Is there a trick or an easy way to show that $M^T$ is the inverse matrix of $M$?
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A transpose of a matrix and its application

Since we know that a matrix can be a tool to describe and change a space,What is exactly a transpose used for and what can we get when we use a transpose to describe a space?Is there a intuitive ...
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Expansion of the Frobenius norm

this might be very elementary question. I was confused by looking at some different sources when expanding the Frobenius norm into trace. Would these two expressions below always be the same? Or only ...
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Proving that the transpose transformation is linear & an iso

Suppose $T: V\rightarrow{W}$ and $T*: W*\rightarrow{V*}$ are linear transformations for finite-dimensional vector spaces, V and W. Note that $T*$ is the transpose. I'm trying to prove that $R: L(V, W)\...
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How to transpose $\left( \frac{\Sigma^{-1} 1 }{1^\top \Sigma^{-1} 1} \right)$? Matrix algebra simplification based on the transpose operator

How does the following matrix algebra reduce dimensionally $$\left( \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}\right)^\top$$ given ...
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Is only one SVD computation enough to perform PCA on a matrix and its transpose?

SVD is used in PCA in order to get the mapping to lower dimensions. Is it enough to perform only one SVD in order to get the PCA for the original matrix AND its transpose, considering that the SVD of ...
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Defining the transpose of a linear map “directly” in terms of the linear map

I am wondering if there is anything wrong with the following way of defining the transpose of a linear map, at least for finite dimensional vector spaces. (The usual definition I'm aware of does it by ...
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What is the transpose of this block/partitioned matrix?

Given matrix $\mathbf{A}^\top \in \mathbb{R}^{nm \times (m+n)}$ has shape $nm \times (m+n)$ and looks like $$\left. \left[ \begin{array}{rrrr|rrrr} 1 & 0 & \dots & 0 & 1 & 0 & ...
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Find transpose of an endomorphism.

Hello everyone I have the following statement : So we have $E$ a vector space with $e_1 , ... , e_n$ as a basis , $\sigma$ a permutation of $ 1,2,...,n$. We are given an endomorphism on $E$ s.t $e_i \...
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If $x$ and $y$ are two linearly independent column $n$-vectors find all the eigenvalues of $xx^{T}-yy^{T}$

If $x$ and $y$ are two linearly independent column $n$-vectors where $n\geq2$ .find all the eigenvalues of $xx^{T}-yy^{T}$
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Edited: Relationship between the eigenvalues of $\frac{1}{2}(A + A^t)$ and $(A^tA)^{1/2}$

Let $A$ be a square matrix and let $A^t$ be its transpose. I would appreciate references to results regarding the relationship between the eigenvalues of $\frac{1}{2}(A+A^t)$ and those of $(A^tA)^{1/2}...
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Finding a matrix A such that T(A) = B [closed]

Given the linear transformation T(A) = A + A^T and B = B^T, find a matrix A such that T(A) = B A should be given in terms of A and B.
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Subscript 1,1 at transponded matrix - what does that mean?

I have to solve a specific equation, but stumbled about a notation I am not sure what it means. I already checked in the internet, but didn't find anything... There's a transposed matrix with a ...
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Intuition behind $x^T A x$

I have been encountering the expression $x^T A x$ (where $x$ is a vector and $A$ is a matrix) in many topics related to linear algebra. However, I fail to grasp the essence; what is happening here? ...
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Transpose of derivative operator acting on Dirac delta

I was reading a text where the transpose $A^T: V^\ast \to V^\ast$ of a linear operator $A: V\to V$ on vector space $V$ was defined as follows $(A^Tf)(v) = f(Av)$ with $f \in V^\ast, v \in V$. (So not ...
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Proving $\mathrm{Rank}A^TA\neq \mathrm{Rank}AA^T$

I am pondering over the question: Whether $\mathrm{rank}\mathbf{AA^T}\stackrel{?}{=}\mathrm{rank}\mathbf{A^TA}\stackrel{?}{=}\mathrm{rank}\mathbf{A}$ for any matrix $\mathbf{A}\in M_{m\times n}$ I ...
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Confusion with definition of transposed matrix using indexed familiy

Background So we have a matrix: $A = (a_{ij}) = \begin{pmatrix} a_{11} & \dots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \dots & a_{mn} \end{pmatrix} \in K^{m \times n}$ The ...
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Can a tensor's indices be flipped arbitrarily?

I made use of the following intermediate step in a demonstration I did: $$...=a_i\sigma_{kj}b_k=a_i\sigma^T_{jk}b_k=...$$ where the "T" just shows that the indices have been flipped. After ...
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Why does this happen: $(v^TA^T)^T\ne vA$

Assuming a matrix $v^TA^T$ is compatible isn't the result of the transpose of their product $(vA)$? In other words, if we assume $v^TA^T$ is compatible. What is the result of $(v^TA^T)^T$? I did the ...

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