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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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?
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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 ...
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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 ...
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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 $$M = \left[\begin{matrix}
M_{1,1} & M_{1,2} & \cdots & M_{1,n}\\
M_{2,1} & M_{2,2} & \cdots & M_{2,n}\\
\vdots & \vdots & \ddots & \vdots\\
M_{n,1} & M_{n,...
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The transpose of an elementary matrix is an elementary matrix of the same type
Here is Prob. 8, Sec. 2.1, in the book _Linear Algebra With Applications by Steven J. Leon and Lisette de Pillis, tenth edition:
Show that if $E$ is an elementary matrix, then $E^T$ is an elementary ...
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Is every symmetric matrix in $\mathbb R^n$ a (non-uniform) stretch?
I conjecture that in $\mathbb R^n$, every symmetric matrix is a non-uniform stretch. Am I correct?
By non-uniform stretch, I mean that if $T$ is a non-uniform stretch, there exists an orthonormal ...
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What does is mean to center variables?
I was reading this answer that is talking about properties of $AA^T$.
If you center columns (variables) of $\bf A$, then $\bf A'A$ is the
scatter (or co-scatter, if to be rigorous) matrix and $\...
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I am stuck in the evaluation of closed form solution to find $\theta$, why $(X^{T}*X)^{-1}*X^{T}*Y$ is used and not $(X*X^{T})$?
$$J(\theta)=\frac{1}{n}\sum_{i=1}^{n} (y^{i}-x^{i}\theta)^2$$
where $y^{i}$ is actual value and $x^{i}\theta$ is predicted for each i.
Differentiating wrt $\theta$ and equating to $0$ (to find $\theta$...
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Transpose of a linear map given by ODE
Consider an ODE
$$
\dot x(t) = A(t) x(t) + B(t) b
$$
$$
x(0) = 0
$$
for $A(t) \in \mathbb{R}^{n\times n}$, $B(t) \in \mathbb{R}^{n\times m}$, $b \in \mathbb{R}^m$
Let's denote by $S(t) : \mathbb{R}^m \...
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Prove $\| A\|_2=\|A^T\|_2$
Let $A \in \mathbb{R}^{n \times n}, $ Prove $\| A\|_2=\|A^T\|_2$
Similarly as the 1-norm I could show that $\| A\|_2=\max_{i} \sqrt{\sum |a_{ij}|^2}$. But this result does not help me to prove this ...
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Dual space and Transposed matrix
Transpose of matrix $T$ denoted by $T'$ is a matrix that holds this: $(T' l , x) = (l, Tx)$ where
$$
T: U \mapsto V ,
x \in U,
$$
and $l$ is in dual of $V$, which is $V'$.
I kinda understand that $(T')...
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Matrix expression of the scalar product between two vectors
My question is in regarding the adjugate matrix in the expression my teacher gave me of the scalar product: $$ \vec{x}\cdot \vec{y}=x^+ G y$$ being $x^+=\overline{x}^T$ the traspose of the conjugate, ...
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Canonical isomorphism between spaces of endomorphisms
In my course notes, given $E$ a $K$-vector space, the canonical isomorphism ($E^{*}$ is the dual space of $E$) between spaces of endomorphisms:
$$
\Phi : End(E) \to End(E^{*})
$$
is constructed via ...
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Determinant of a matrix is equal to the determinant of its transpose
For a square matrix A, I want to show that $$ | \mathbf{A}^T| = |\mathbf{A}| $$
Proof:
Let the same permutation that changes $ \varphi(j_1,...,j_n) $ into $ \varphi(1,...,n) $, change $ \varphi(1,...,...
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Why a matrix has to be invertible for $\operatorname{adj} A^T=(\operatorname{adj}A)^T$ to be true?
I read a theorem
If $A$ is an invertible square matrix, then $\operatorname{adj} A^T= (\operatorname{adj} A)^T$.
But after attempting to prove it myself and also reading the proof I am unable to ...
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Dot product between two vectors transformed by orthogonal matrices
I am reading through the "Matrix Transformations" chapter of this book and more specifically on Orthogonal Matrices. I understand their properties and understand that multiplying vectors by ...
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Understanding the dimension of a vector?
In my research work, the dimension of $\textbf{y}$ is $N\times 1$, where $\textbf{y}$ has both real and imaginary parts.
So the dimension of real part of $\textbf{y}^T$ i.e., $\mathcal{R}(\textbf{y}^T)...
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When is $A^TA$ invertible?
Let $A$ be a matrix. What are some necessary/sufficient conditions for the Gram matrix $A^T A$ to be invertible?
This question came up when I was trying to learn about least-squares regression. Is it ...
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Is $1/2(A+A^T)$ always elementary given that $A$ is elementary.
Can we conclude that $$1/2(A+A^T)$$
is always elementary given that A is elementary and $A\in \mathbb{M_{2\times2}(C)}$
I have tried proving it using the ...
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For a square matrix $A$ over any field, does $A^TA=0$ imply $AA^T=0$?
If $A^TA=cI$ for some scalar $c\neq0$, then $A^T=cA^{-1}$, and thus $AA^T=cI$ as well. (The first equation says that $c^{-1}A^T$ is a left-inverse of $A$. And a left-inverse is also a right-inverse: ...
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Matrix made up by $2$ orthonormal vectors $x$ and $y$ in $\mathbb{R^2}$ and $y$ is transposed
I came across the following exercise from my professor and had no idea how to start. I tried it with the standard basis in $\mathbb{R}^{2}$, which results in a rank of $1$ and eigenvalue $0$, but this ...
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Is $\mathbf{A}\mathbf{A}^T$ symmetric? [duplicate]
Is matrix
$$\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 ...
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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 ...
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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 ...
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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 ...
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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$. ...
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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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What effect does taking the transpose of a linear transformation matrix have on the transformed plane
I have recently been using 3Blue1Brown's fantastic YouTube series on linear algebra to supplement my third year matrix analysis course. I have found it super useful to be able to visualise matrices as ...
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Using Integration by Parts for Transpose of a Derivative
In Introduction to Linear Algebra: Fifth Edition by Gilbert Strang, it supposes that we have a matrix $A$ such that it differentiates the elements in a vector $x$ via $Ax$. It then resolves to find ...
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Prove that $\operatorname{adj}(A^t)=(\operatorname{adj}(A))^t$
I know this theorem is correct but I'm struggling to prove it formally:
For any square matrix A, $\operatorname{adj}(A^t)=(\operatorname{adj}(A))^t$.
Any explanation would be greatly appreciated.
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SVD of matrix and SVD of its transpose: U and V exchange places
I have reading the post of SVD of matrix vs. SVD of its transpose for m < n
I am not sure whether I agree with his conclusions:
So, let $\mathbf{A}(m \times n)$ and $\mathbf{B}(m \times n)$ be two ...
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Order of transposes for matrix with matrix elements?
So say I have a matrix
$$\begin{bmatrix}
\begin{bmatrix}
1 \\ 2
\end{bmatrix}^T
&
\begin{bmatrix}
3 \\ 4
\end{bmatrix}^T
&
\begin{bmatrix}
5 \\ 6
\end{bmatrix}^T
\end{bmatrix}^T.$$
How would ...
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Substitution in a matrix
I have been trying to understand this solution and I can not wrap my head around the fact that $a_{11}$ = $a_{22}$ = 0. What is this conclusion based on? Is it due to the fact that they are related by ...
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Exponential of symmetrical 4x4 matrix [closed]
Consider a matrix $A$, such that $A_{ij}=A_{ji}$
I want to find $e^{itA}$
I tried to represent $A$ as sum of 10 different matrices, that would show the symmetry, but the final result ends up to be ...
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Exercise 7, Section 3.7 of Hoffman’s Linear Algebra
Let $V$ be a finite-dimensional vector space over the field $F$. Show that $T \to T^t$ is an isomorphism of $L(V, V)$ onto $L(V^*, V^*)$.
Note: For $T\in L(V, W) $ , the dual map (or transpose) $T^t\...
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Transpose of automorphisms
I have recently come across the result that ${\rm Aut}(\mathbb{Z}^n, +)$ is isomorphic to $GL_{n}(\mathbb{Z})$. Since transposition does not alter the determinant of a matrix, $A^{T} \in GL_{n}(\...
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Motivation for the transpose map
This post has some answers that give some intuition as to the definition of the transpose. My rudimentary (perhaps inaccurate) understanding is that for a linear transformation $T: V \to W$, we're ...
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Does $b^Tx = x^Tb$ always hold?
Does $b^Tx = x^Tb$ always hold?
(assume that $b^Tx$ can be defined )
I am studying some computer science and it seems like this property holds for transpose matrices but I am not sure. I found that :
...
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Computing $D_v^2(f)(x_0)$, if $f(x,y)=x^3+x+y^2-2y+1$, $v=\frac{1}{5}(3,4), x_0=(\frac{1}{3},1)$
Compute $D_v^2(f)(x_0)$, if $f(x,y)=x^3+x+y^2-2y+1$, $v=\frac{1}{5}(3,4), x_0=(\frac{1}{3},1)$ using the formula $D_v^2(f)(x_0)=v^TH_f(x_0)v$.
So, $v^T= \left[
\begin{array}{cc}
\frac{3}{5}\\
\...
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How to show $(Ax)^T = x^T A^T$ for square matrix $A$?
I have vector $x \in \mathbb{R}^n$ and square matrix $A \in \mathbb{R}^{n \times n}$ (doesn't depend on $x$). I believe I can make the claim $(Ax)^T = x^T A^T$.
On paper with $n = 2$, I can calculate:...
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How to scale a fat matrix to ensure its rows are orthonormal?
I have the following matrix
$${\bf B} = \begin{bmatrix}
0&-4.2423&4.2423&1.4871\\
1.6532&-1.2735& -1.2735&0.0024\\
0 & -0.2805 & 0.2805 & -0.8823
\end{bmatrix}$$
I ...
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Finding the conjugate/hermitian transpose of a transformation
Hopefully, I'm using the correct terms/names of things, mainly because the language in which I study is not English.
Given the operator $T$, in this case is the derivative operator , with the inner ...
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Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric
In a linear algebra textbook, I was given the following problem:
If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric.
I know that there are relatively ...
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Matrix of adjoint linear map is transposed matrix
I am stuck with some exercise and hope that someone can give me a hint:
Let $V$ and $W$ be vector spaces and let $B_V = \big\{ v_1, \ldots, v_n \big\}$ and $B_w = \big\{ w_1, \ldots, w_n \big\}$ be ...
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When is a sum of products of two matrices and the transposes positive definite?
Let $X$ and $Y$ be $n \times m$ matrices. The matrix
$$
A = X^TY + Y^TX
$$
will be a $m \times m$ square, symmetric matrix. Is it possible to say: i) when is $A$ is positive definite? and when it is ...
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Are left and right operator norms equal?
If $V$ is a finite dimensional normed vector space and I identify row and column vectors then I can define the action of a matrix both from the left and from the right. I can define the left and ...
