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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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Transpose of rational matrix is also rational

Rational numbers can be defined as those numbers $a \in \mathbb{R}$ for which there exists an integer $v \in \mathbb{Z}$ such that $av \in \mathbb{Z}$. Let us consider the following higher-dimensional ...
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Equation with Exponential matrix

I have the following equation: \begin{gather} (e^{At})^T \cdot P \cdot e^{At} - P = - I \end{gather} where im trying to find the matrix $P$. Is there any name to this equation or some simple ...
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if $x,h \in \mathbb{R}^d$ and $A \in \mathbb{R}^{d\times d}$ is it possible to justify that $(x^TAh)^T = h^TA^Tx$? [duplicate]

if $x,h \in \mathbb{R}^d$ and $A \in \mathbb{R}^{d\times d}$ is it possible to justify that $(x^TAh)^T = h^TA^Tx$?
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Matrix multiplication $AB=0$ so $A=0 $ or $B=0$

I have tried with the idea of $AA^T=0 $ and use the trace, but nothing changed. I have also to prove : if $AB=0 $ then $BA=0$ .
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Prove if a matrix A multiplied by its transpose is 0, then the matrix A is nule

if A is a matrix NxN prove that if A x A^t = matrix nule NxN so A is nule NxN I'Have tried by the summation notation but nothing came
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Are the generalized $\lambda$ eigenvalues of a matrix T and its transpose the same?

I know that the eigenvalues of a matrix $T$ and its transpose $T^T$are the same, but is this true for the generalized eigenvalues? i.e. if a matrix $T$ has $\lambda$ as a generalized eigenvalue, is ...
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Prove that $\det(A A^t) =0$ for every $4 \times 3$ matrix $A$

I try to prove that the determinant of $A A^t$ equals $0$. $A^t=(A \text{ transpose})$. I tried to prove it by contradiction. if $\det(A A^t )\neq 0$ so there exists an inverse, so I tried to ...
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Differentiation of a Matrix transpose

So basically I wanted to figure out why the differential of the following expression is what it is: \begin{align} S &= e'e \\ &= (y-W\beta)'(y-W\beta) \\ &= \underbrace{y'}_{1\times T} \...
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How to prove the transpose matrix is in a vector space with restrictions on the dimension

For an assignment in class, I have the following question. Let n $\geq 1$ and let W be a subspace of $Mn\times n(K)$ such that $dim(W)>\frac{n^2-n}{2}$. Prove that W contains a non-zero matrix ...
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upper bound for norm features inequality with arbitrary vector

Let $Q$ be a positive definite matrix and $w$ be an arbitrary vector. How we can conclude about upper bound of $ww^{T}$ if $w^T Q^{-1} w \leq 1$?
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What is the meaning of $A^T A$?

Given are an $m$-dimensional vector-space $V$ and a base $\mathscr{B}$ of $m$ vectors $\mathbf{e}_i \in V$. Given $m$ vectors $\mathbf{a}_j$ of an $n$-dimensional vector-space $W$, one can construct ...
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Why is $A^T : W^* \to V^*$ if $A : V \to W$?

I was wondering what the meaning of a transpose matrix was. I found an answer on this forum, but there's something I don't understand. It was said that if $A$ transforms $\mathbf{x}$ into $\mathbf{y}$...
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Why is this notation equal to its transpose?

In my econometrics textbook, I have this step which is not clear to me: \begin{align} S &= e'e \\ &= (y-W\beta)'(y-W\beta) \\ &= \underbrace{y'}_{1\times T} \ \underbrace{y}_{T\times 1} ...
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Is there a metric that measures similarity of two complex fields / matrices

I have two $a \times b$ matrices of complex numbers, representing acoustic fields at some planes, and I want to measure their similarity. One way I know is the cosine similarity, but the cosine ...
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vector transpose multiplied by matrix then multiplied again by the same vector, find vector

Suppose w is a 1-d vector, X is a d by d positive semi-definite symmetric matrix, and a is a constant. Is there a way to express w in terms of X, a? $$ w^T X w = a^2 $$ I was thinking something like $...
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Relation between eigenvalues of $A^{\top}BB^{\top}A$ and $B^{\top}AA^{\top}B$

I have two real value matrices: $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{m \times p}$. If I know which are the eigenvalues of $A^{\top}BB^{\top}A$, what can I say about the eigenvalues ...
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What is the angle between a $A_{3x3}$ with $Rank(A)=2$ and $A^T$

I need to solve this problem: For Matrix $A_{3x3}$ with $Rank(A)=2$. If Matrix A is Transposed and its elements are the same as elements of Matrix B. What is the angle of rotation from A to B?
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Prove that $\phi(f(X),Y)=\phi(X,f(Y))~\forall X,Y\in\mathbb R^3$ where $\phi(X,Y)=X^TAY$ and $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$

Within this AoPS thread there was the following question asked Let $\phi(X,Y)=X^TAY$ be a scalar product on $\mathbb R^3,$ and let $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$, where $$A=\begin{...
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Are there more matrix transpose properties than this one?

In a recent exercise, our professor wrote that $t^TXw = w^T X^T t$ where $t,w$ are vectors and $X$ is a matrix. I tried it for a simple example and the identity seems to hold true. However, I can't ...
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Does the concept of “adjoint map” determine the metric up to scaling?

Let $V$ be a real finite-dimensional vector space, and $g$ and inner product on $V$. $g$ induces a concept of "adjoint map" , i.e. a linear map $\text{Hom}(V,V) \to \text{Hom}(V,V)$ given by $S \to S^...
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Is the matricial representation of the inverse relation the transpose of the representation of the original relation?

here we are studying relations on AxA with A a finite set. We represent the relation on a matrix in this way: $M\left ( R \right )_{ij}=1 \; if (i,j) \in R; and M\left ( R \right )_{ij}=0 \;\;...
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What are the eigenspaces and the minimal polynomial of the “transposing about the anti-diagonal” endomorphism?

Given a field $K$ of characteristics zero, let the endomorphism $f:M_2(K)\to M_2(K)$ be defined by $f\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=\begin{bmatrix}d&b\\c&a\end{...
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Geometric Relationship between Two Vectors [duplicate]

Consider two column vectors such that $a = (1,2,3)^T$ and $b = (-3,3,-1)^T$. What is the geometric relationship between $a$ and $b$?
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Diagonalizability in relation to squaring and transposition

True or False? Let A be a square matrix If $A$ is diagonalizable, then $A^2$ is diagonalizable. If $A$ is diagonalizable, then $A^t$ is diagonalizable. Re 1, my answer is that it is correct, but I ...
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How to prove that $(AB)^t = B^tA^t$

The proof given in my book (and I came up with as well) is: However, the part that throws me off is line #3 where they do $\Sigma A_{jk} B_{ki} = \Sigma B_{ki} A_{jk}$ I understand that ...
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Transpose of $Y ={A}^{*} \operatorname{diag} \left( b \right) {A} $

What would the tranpose of $Y$ look like $$Y ={A}^{*} \operatorname{diag} \left( b \right) {A} $$ where $*$ is the conjugate tranpose, i.e., hermitian.
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Transpose rules and algebra

A, B and P are all squared matrices of the same order ($nXn$). It is given that: $PP^{T}=I$ $B^{T}B=I$ $A=P^{T}BP$ Which of the following is correct: $(1) A=B$ $(2) AA^{T}=I$ $(3) (PB)^{-1}=(...
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how does sqrt(n)/n equal 1/sqrt(n)?

I'm working through a probability book where the following is stated (chapter 3): Definition 3.3 Let $a_n$ and $b_n$ be two sequences of numbers. We say $a_n$ is asymptotically equal to $b_n$, ...
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If $x$ and $y\in\Bbb{R}^{n}$ are eigenvectors for $\lambda\neq\mu$, respectively, show $x^{T}\cdotp y = 0$

For $x^{T}\cdotp y = 0$, I understand that I can either look at it through matrix multiplication $x^{T}y^{T} = 0$ as you can't do that multiplication. I'm very sure this isn't the right way of looking ...
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Transpose notation

After using the transpose for a while, I wondered if there in any vague connection to other superscript stuff like exponents or something. I doubt it, but I couldn't find anything and it seems weird ...
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If $A=A^2$ is then $A^T A = A$?

I know that for a matrix $A$: If $A^TA = A$ then $A=A^2$ but is it if and only if? I mean: is this true that "If $A=A^2$ then $A^TA = A$"?
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Compare classical and modified algorithm of Cholesky

Let $\textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $\textbf{A} = \textbf{R}^T\textbf{D}\textbf{R}$ and the classical factorization $\textbf{A} = \textbf{R}_c^T\textbf{R}...
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Rank, nullity and the number of rows of a matrix

I have this question here: Let $A$ be a matrix with $10$ columns, $dim$ $Null(A)=5$ and $dim$ $Null(A^T)=3$. How many rows does $A$ have? $a)$ $8$ $b)$ $3$ $c)$ $5$ $d)$ $10$ ...
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The Derivative of a Vector

It is stated in this video that some books attempt to define the derivative of a vector, with the added caveat that tensor notation is a much better approach. Presumably, this is a different concept ...
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Derivative of a transpose variable [duplicate]

I'm trying to solve the derivative of an equation that has a transpose (calculation of a multivariate distribution : trying to calculate the mean) d/dx (x(T)Ax), where x is a vector, x(T) is the ...
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A matrix is similar to its transpose without jordan form

Let $A \in M_{n×n}\mathbb{(C)}$ be invertible. Then $(A^∗)^{−1} = (A^{−1})^∗$ Let B be a nonsingular matrix such that $A = B^{−1}B^∗$ . Show that $A^{−1}$ is similar to $A^∗$. ($A^*$ is ...
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Let ${^ts}:E\to F$ be the transpose of $s:F\to E$. Show that $\text{Im}(s)\cap \ker (\,^ts)=\{0_E\}$.

Let $s\in \mathcal{L}(F,E)$ $$\displaystyle F \overset{s}{\longrightarrow} E\overset{^ts}{\longrightarrow} F$$ I spent two days to show that :$$\text{Im}(s)\cap \ker (\,^ts)=\{0_E\}\qquad \tag{...
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Matrix-vector product: representing matrix as vector of vectors seemingly leads to paradox when transposing the matrix

I'm currently taking a university class on linear algebra. In some proofs, a given matrix $A \in \mathbb{R}^{m\times n}$ is said to be able to be represented as a $1\times n$ row vector of $m \times 1$...
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Inverse $4\times 4$ matrix contains left upper $3\times3$ transposed matrix

I am not a matrix geek or something. I just remeber a couple of things from the university math classes. Maybe the explanation is simple. What's going on: I go through a code of a certain game and an ...
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I know symmetric matrix $S = QDQ^T$, but how can matrices with form ADA be symmetric?

I have learned that a symmetric matrix must be able to be written in form of $S=QDQ^T$ where Q is the orthonormal eigenvectors. But I saw an example that display a symmetric matrix in the form of $S = ...
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Is it possible to transpose a square matrix by multiplication?

Is it possible to transpose matrix by left and/or right multiplication? $XAY = A^T$ Do $X$ and $Y$ exist for any $A$?
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Stuck on a proof about rotating a matrix

Given a matrix $A \in M^{n×n}(F)$, let $A^{\rho}$ denote the matrix obtained from $A$ by ‘rotating’ it $90^{\circ}$ clockwise. For example, $$\begin{bmatrix} 1&2\\ 3&4 \end{bmatrix}^\rho =\...
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General chain rule - transpose

There is a function: $$g_{(x)}:\mathbb{R}^n \rightarrow \mathbb{R}$$ The $1_{st}$ derivative is gradient, $\mathbb{R}^{n}$ column vector. Second derivative is Hessian, $\mathbb{R}^{n*n}$ matrix. ...
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How to calculate missing components of rotation matrix?

Q=\begin{bmatrix}1/\sqrt3&1/\sqrt3&1/\sqrt3\\ Q21&Q22&Q23\\Q31&Q32&1/\sqrt2\end{bmatrix} \sqrt
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if $A+A^T$ is stable, then is $A$ also stable?

Let $A \in \mathbb{R}^{n\times n}$ such that $A+A^T$ has stable eigenvalues (their real part is strictly negative). I remember seeing somewhere that this involves that $A$ is also stable but I do not ...
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Find all square matrices $A$ satisfying ${}^t\!A = -A$.

I am given an $n \times n$ matrix $A$. If $A = A^2$ and ${}^t\!A = -A$, I need to find matrix of $A$. $$A^2=A⟹A^2⋅{}^t\!A=A⋅{}^t\!A⟹A^2⋅(-A)=A⋅(-A)$$ How do I show the matrix of A from here ?
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Fail to get original term regarding Multivariate Gaussian Distribution

I have a project regrading Robotics. I want to implement Extended information filter. But today my query is not about Robotics Topic. It is mathematical based doubts. The key formula in which Extended ...
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$AA^T = BB^T \rightarrow A =BU $

Is it true that $AA^T = BB^T$ always implies $A =BU $ for some unitary matrix $U$. Or there may exist other scenarios also? All matrices are real and square.
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Derivate of vector : transpose, conjugate and conjugate transpose

Let $x$ and $y\in \mathbb{C}^{K\times 1}$ and $H\in \mathbb{C}^{K\times K}$ a diagonal matrix. $\bar{x}$ denotes the complex conjugated, $x^{T}$ denotes the transpose and $x^{*}$ denotes the complex ...
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Transpose the answer matrix to the right dimension after partial differentiation the product of matrices

I would like to understand about transposing matrix after partial differentiation for the right dimension of answer For a function of the product of matrices $W$ and $H$ with dimension $i*k$ and $k*j$...