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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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Proof about Hadamard matrices that $H H^\textsf{T} = n I_n$ [closed]

I have read the Wikipedia article about Hadamard matrices that says: Let $H$ be a Hadamard matrix of order $n$, the following is true: $H H^\textsf{T} = n I_n$, where $I_n$ is the identity $n×n$ ...
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Product of Conjugate Transpose with Itself equalling the Zero Matrix. [duplicate]

I am reading a linear algebra book and I am stuck on one of the questions that asks to (a) : Find all nxn matrices with real entries such that $$ A^{T}A = 0 $$ (b) : Find all nxn matrices with complex ...
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Find a matrix $Q$ that has $Q(Q^T)=I$ but does not have orthonormal columns

My textbook says that if a matrix $Q$ is square and has orthonormal columns then $Q(Q^T)=I$, but it does not say the opposite (that if $Q(Q^T)=I$ then $Q$ has orthonormal columns). Is there an example ...
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Why $\nabla f$ do not exactly coincide with $D f$ (it's its transpose)

Is there any reason (historical, or of any other kind) to why $$\nabla f= \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \\ \end{bmatrix}...
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Matrix algebra: transpose and inverse of product

In this video about Kalman filters, video here, the lecturer at 42:00 describes the Schwarz inequality as $$Q^\top Q \geq (P^\top Q)^\top(P^\top P)^{-1}(P^\top Q)$$ for nonsingular $P^\top P$ but ...
insipidintegrator's user avatar
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If $B$ is a $n \times n$ matrix, then is the column space of $B$ and $BB^{'}$ same?

The question: If $B$ is a $n \times n$ matrix, then is the column space of $B$ and $BB^{'}$ same, where $B^{'}$ is transpose of $B$? Context: There was a question that if $ABB^{'} = CBB^{'}$, then ...
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Solving y = BX using Gram Matrix

To find regression coefficients in linear regression, BLUE estimator in particular, the formula seen below is a good estimator: $$ B = (X^T X ) ^{-1}X^T y $$ In this setting, y and X are known and B ...
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Transform a column vector into its transpose with matrix multiplication? [closed]

I don't have tensor in mind here, what follows is just a question about linear algebra: If I start out with an $n\times 1$ column vector over the reals, are there any combinations of matrix ...
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What is the matrix representation of the derivative of the vector transpose

From the definition of the Fréchet derivative and the linearity of the transpose it’s clear that the derivative of the vector transpose is the vector transpose itself. $$f(x + h) = f(x) + D(x)h + o(\...
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How to show that the determinant of a square matrix is equal to the determinant of its transpose?

Consider an $n\times n$ square matrix A. Let $B = A^{T}$ We know that $|B|$ is the sum of n-component products $b_{1i}b_{2j}....b_{nr}$ where $(i, j, ..., r)$ is a permutation of the natural order $(1,...
Vibhor Verma's user avatar
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Prove the following matrix equation: $A'B = B'A$

Let's say I have these 2 matrices: $$A = \begin{bmatrix} c \\ d \end{bmatrix} $$ and $$B = \begin{bmatrix} e \\ f \end{bmatrix}$$ $A'B = ce + df $ and $B'A = ec + fd$ As shown above, $A'B = B'A$. But ...
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Which item do we transpose for the dot product?

I cannot find anywhere whether it's a rule to always transpose the first item in a dot product equation. The example I have is for 2 column vectors (a and b) both have 3 rows. Because the order ...
Max's user avatar
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Adjugate vs transpose of a matrix

Adjugate(classical adjoint) of a matrix https://en.m.wikipedia.org/wiki/Adjugate_matrix is the inverse rotation without scaling. When a matrix is multipled by its inverse, off diagonal entries are ...
Jay's user avatar
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Changing sign of number when transposing [closed]

In an equation, when we change a variable or number sides, we change its sign from positive to negative and vice-versa, right? So in 738 = a + b, shouldn't it be: <...
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Prove the geometric nature of matrix transpose: same stretch, inverse twist

Background & Motivation The geometric and intuitive nature of matrix transpose is well explained (e.g. What is the geometric interpretation of the transpose? and Truly intuitive geometric ...
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Tranpose of a mixed tensor

If $(-)'$ denotes the transpose operation then for a type (1,1) mixed tensor ${T^a}_b$, $\left({T^a}_b\right)'={T^b}_a$. Although this seems right I would like to be able to show this starting with a ...
Ted Black's user avatar
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Can we prove $A \hat x = r \hat y$ implies $A^T \hat y = r \hat x$?

Let $A$ be a real matrix. If $A \hat x = r \hat y$, does $A^T \hat y = r \hat x$? I believe it does, but am struggling to prove it. Geometrically, I believe this is true, because $r$ captures the ...
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Orthogonal basis but not unitary matrix inverse?

For an orthgonal matrix, which has orthogonal+normalized sets of basis, it is said that $A^T$=$A^-1$. But what about matrix with orthogonal but not normalized set of basis? Does it have some kind of ...
Lime nut's user avatar
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If $ A $ and $ A^T $ commute, do they share the same eigenvectors when they are not diagonalizable?

I saw this on Strong's book Introduction to Linear Algebra, but it didn't prove it: If $ A $ and $ A^T $ commute, then they share the same eigenvectors. And now I am wondering whether it is still ...
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"Simple" proof that an n x n matrix is similar to its transpose

At the risk of getting voted down for duplicating A matrix is similar to its transpose or Proving an $n\times n$ matrix is similar to its transpose, or such, I didn't feel comfortable "answering&...
Blue Ghost's user avatar
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Does this condition imply orthogonality?

In transforming matrices between bases I have come across this curious equation: $\mathbf{A} \mathbf{B} \mathbf{C} = \mathbf{A}^{\mathsf{T}} \mathbf{B} \mathbf{C}^{\mathsf{T}}$, for all $\mathbf{B}$. ...
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How to prove this matrix inequality?

Let $A$ be an $n\times m$ real matrix, and let $B$ be an $n\times(n-m)$ real matrix, please prove this matrix inequality. $A^{\prime}$ represents the transpose of the matrix. $$ \det\begin{bmatrix} A^...
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$A^{-}A$ and $AA^{-}$ are symmetric when $A^{-}$ is reflexive

I'm currently trying to solve the following statement regarding generalized inverse matrix $A^{-}$; If $A^{-}$ is reflexive, then $(A^{-}A)' = A^{-}A$ and $(AA^{-})' = AA^{-}$ To start with, $A^{-}$ ...
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suppose $A$ is an $m\times n$ real matrix and $A^\top A$ is $3\times3$ nonsingular matrix, then $n =$? [closed]

the answer of this question is $n<3$ , why? I mean how we decided that is $n<3$ when $A$ multiply $A^\top$ will be singular?
Lina Naif's user avatar
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Z-matrix multiplied by its transpose

What can be said about eigenvalues and eigenvectors of $ZZ^T$ if Z is a Z-matrix whose columns sum to zero?
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What does the transpose of a linear transformation represent? [duplicate]

I'm struggling to understand what the transpose of a linear transformation represents. My textbook's motivation for this wasn't very helpful. All it did was ask, "Is there a linear transformation ...
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Prove if $M^2 =0$, then $\operatorname{rank}(M+M^T)=2\operatorname{rank}(M)$.

Prove if $M^2 =0$, then $\operatorname{rank}(M+M^T)=2\operatorname{rank}(M)$. I have used the rank nullity theorem and the fact that rank of matrix and its transpose is same to prove it but I am not ...
Black Panther's user avatar
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My calulations show that for all $n\times m$ matrices $A, B$, it is $det(A^tB)=0$. Where is the mistake?

Let $A, B \in \mathbb{M}^{n\times m}(\mathbb{R})$ be two $n\times m$ real matrices. Then the product $A^t\cdot B$ is a $m\times m$ real matrix, thus, it has a determinant $D=det(A^t B)$. I calculated $...
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What's the result of $(A^{-1}BC^{T})^{T}$?

I know it's a very simple one but still worth it, say we have 3 real matrices 3x3, $A,B,C$, what is $(A^{-1}BC^T)^T$? Is it $(A^{-1})^T B^T C$? Or since it is transpose we have to change the order of ...
Acedium 20's user avatar
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Transposing a matrix using matrix multiplication

I came across an interesting problem in a linear algebra problem book. This is the very first paragraph in the problem book, it deals with basic operations with matrices: multiplication by a number, ...
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Invertibility of Block Matrix Partial Transpose

Let $$M = \left[\begin{matrix} M_{1,1} & M_{1,2} & \cdots & M_{1,n}\\ M_{2,1} & M_{2,2} & \cdots & M_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ M_{n,1} & M_{n,...
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Is every symmetric matrix in $\mathbb R^n$ a (non-uniform) stretch?

I conjecture that in $\mathbb R^n$, every symmetric matrix is a non-uniform stretch. Am I correct? By non-uniform stretch, I mean that if $T$ is a non-uniform stretch, there exists an orthonormal ...
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I am stuck in the evaluation of closed form solution to find $\theta$, why $(X^{T}*X)^{-1}*X^{T}*Y$ is used and not $(X*X^{T})$?

$$J(\theta)=\frac{1}{n}\sum_{i=1}^{n} (y^{i}-x^{i}\theta)^2$$ where $y^{i}$ is actual value and $x^{i}\theta$ is predicted for each i. Differentiating wrt $\theta$ and equating to $0$ (to find $\theta$...
Tipu Sultan's user avatar
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Transpose of a linear map given by ODE

Consider an ODE $$ \dot x(t) = A(t) x(t) + B(t) b $$ $$ x(0) = 0 $$ for $A(t) \in \mathbb{R}^{n\times n}$, $B(t) \in \mathbb{R}^{n\times m}$, $b \in \mathbb{R}^m$ Let's denote by $S(t) : \mathbb{R}^m \...
tom's user avatar
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Prove $\| A\|_2=\|A^T\|_2$

Let $A \in \mathbb{R}^{n \times n}, $ Prove $\| A\|_2=\|A^T\|_2$ Similarly as the 1-norm I could show that $\| A\|_2=\max_{i} \sqrt{\sum |a_{ij}|^2}$. But this result does not help me to prove this ...
piero's user avatar
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Dual space and Transposed matrix

Transpose of matrix $T$ denoted by $T'$ is a matrix that holds this: $(T' l , x) = (l, Tx)$ where $$ T: U \mapsto V , x \in U, $$ and $l$ is in dual of $V$, which is $V'$. I kinda understand that $(T')...
amineh's user avatar
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Matrix expression of the scalar product between two vectors

My question is in regarding the adjugate matrix in the expression my teacher gave me of the scalar product: $$ \vec{x}\cdot \vec{y}=x^+ G y$$ being $x^+=\overline{x}^T$ the traspose of the conjugate, ...
Aley20's user avatar
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Canonical isomorphism between spaces of endomorphisms

In my course notes, given $E$ a $K$-vector space, the canonical isomorphism ($E^{*}$ is the dual space of $E$) between spaces of endomorphisms: $$ \Phi : End(E) \to End(E^{*}) $$ is constructed via ...
Ta Thanh Dinh's user avatar
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Determinant of a matrix is equal to the determinant of its transpose

For a square matrix A, I want to show that $$ | \mathbf{A}^T| = |\mathbf{A}| $$ Proof: Let the same permutation that changes $ \varphi(j_1,...,j_n) $ into $ \varphi(1,...,n) $, change $ \varphi(1,...,...
Fledermaus's user avatar
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Why a matrix has to be invertible for $\operatorname{adj} A^T=(\operatorname{adj}A)^T$ to be true?

I read a theorem If $A$ is an invertible square matrix, then $\operatorname{adj} A^T= (\operatorname{adj} A)^T$. But after attempting to prove it myself and also reading the proof I am unable to ...
Ayush Upadhyay's user avatar
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Dot product between two vectors transformed by orthogonal matrices

I am reading through the "Matrix Transformations" chapter of this book and more specifically on Orthogonal Matrices. I understand their properties and understand that multiplying vectors by ...
Georgi B. Nikolov's user avatar
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When is $A^TA$ invertible?

Let $A$ be a matrix. What are some necessary/sufficient conditions for the Gram matrix $A^T A$ to be invertible? This question came up when I was trying to learn about least-squares regression. Is it ...
a_____'s user avatar
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For a square matrix $A$ over any field, does $A^TA=0$ imply $AA^T=0$?

If $A^TA=cI$ for some scalar $c\neq0$, then $A^T=cA^{-1}$, and thus $AA^T=cI$ as well. (The first equation says that $c^{-1}A^T$ is a left-inverse of $A$. And a left-inverse is also a right-inverse: ...
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Matrix made up by $2$ orthonormal vectors $x$ and $y$ in $\mathbb{R^2}$ and $y$ is transposed

I came across the following exercise from my professor and had no idea how to start. I tried it with the standard basis in $\mathbb{R}^{2}$, which results in a rank of $1$ and eigenvalue $0$, but this ...
Maxim Decherf's user avatar
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Is $\mathbf{A}\mathbf{A}^T$ symmetric? [duplicate]

Is matrix $$\mathbf{B}=\mathbf{A}\mathbf{A}^T$$ necessarily symmetric (i.e., does $\mathbf{B}=\mathbf{B}^T$)? Writing $$b_{ij}=a_{ij}a_{ji}$$ seems wrong because the $j$ on the LHS is different from ...
Geremia's user avatar
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Intuition behind thinking of the transpose of matrices?

Not directly related to the title, but I recently went through LADR, and did not find it particularly intuitive. I've started to work through LADW. I've found it much more intuitive. There is one ...
Frank Nakasako's user avatar
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Solve for vector $r$ in an equation involving sum of the square elements of $r$ and the sum of the square elements of $Mr$ where $M$ is a known matrix

Hello stackexchange community. I have a system of equations where the aim is finding two vectors $r$ and $c$ of lengths $t$ and $m$ respectively. There is a known matrix $M$ of shape $m\times t$ and ...
Sumanyu Asthana's user avatar
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Conceptualizing and Contextualizing Quadratic Forms

In an 'Estimation Theory' course that I'm taking, I notice over and over that the equations we work with are littered with what I think are called 'Quadratic Forms'. In case that terminology is ...
David D.'s user avatar
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Is congruency same as $*$-congruency for real matrices?

Two real matrices $A,B\in M_n(\mathbb R)$ are said to be congruent to each other if there exists an invertible matrix $P\in GL_n(\mathbb R)$ such that $B=P^tAP$, where $P^t$ is the transpose of $P$. ...
sagnik chakraborty's user avatar
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Knowing solution of ${\bf A x} = {\bf b}$, find solution for ${\bf A}^\top {\bf x} = {\bf b}$ [closed]

Suppose that ${\bf A} \in \Bbb R^{n \times n}$ is invertible, ${\bf b} \in \Bbb R^{n}$. Knowing a solution of ${\bf A} {\bf x}_1 = {\bf b}$, find a solution for ${\bf A}^\top {\bf x}_2 = {\bf b}$. ...
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