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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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 ...
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Does $A=B$ imply that the rank of $A$ equals the rank of $B$?

I am trying to understand a poof of the clubs of oddtown theorem, and I am stuck at this step: If $A$ is an $m \times n$ matrix and $AA^T=I_m$, then the rank of $AA^T$ is AT LEAST $m$. Now if $A=B$ ...
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Show that, in general, $A^TA\neq AA^T$.

Suppose that $A$ is a real square matrix with transpose $A^T$. I am trying to show that, in general, $$A^TA\neq AA^T.$$ As an example, consider the matrix $A=\begin{pmatrix} 1 & 0 \\ ...
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rank of a matrix and its transpose

Let $A$ is a $n \times n$ matrix, and $B$ is a $n \times m $ matrix and $m\leq n$. If $rank[A\quad B]=n$, can we get $rank[A\quad B*B']=n$? I try to prove that $B$ and $B*B'$ have the same range space....
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When do we have $(AB)^T= AB$?

When is it true that $(AB)^T= AB$? I'd say that if $A$ commutes with $B$ it is true, but i don't know if there is any other case where this is true.
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Are there conditions where $\pmb{ABA}^{-1}=\pmb B$ when $\pmb A$ and $\pmb B$ do not commute?

Given that $\pmb{AB} \ne\pmb{BA}$ and $\pmb{BA}^{-1} \ne\pmb A^{-1}\pmb B$, are there conditions where $\pmb{ABA}^{-1}=\pmb B$ ? I am working in the context of a continuous, LTI system $\pmb{\dot x}(t)...
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If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric? I know that given the symmetric matrices $A$ and $B$, then $...
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Gramian of product with diagonal matrix plus outer product

The Gramian of a matrix $M$ is $\det M^\dagger M$. Let $A = \operatorname{diag} a + b \otimes c$. Under what conditions can the Gramian of $A B$ be simplified and expressed in terms of the Gramian of $...
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Is the transpose of $T$ also the dual map of $T$?

Suppose I have $T: V \to W$. Then $T$ defines a map $T': W' \to V'$ that some sources (e.g. Axler) call the dual map of $T$. And $T$ defines a map $T^t: W' \to V'$ that some sources (e.g. Hoffman and ...
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Simple Matrix Equation.

$$(2\begin{bmatrix}2 &1 \\-1 &3\end{bmatrix} - 5A^{-1})^{T} = (4A^{T})^{-1}$$ I have approached this question by inverting the transpose and inverse operation on the LHS and then distributing ...
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Meaning of left multiplying a matrix A with another B and then right multiply the result with a transposed B?

I am currently in the process of learning Kalman Filters and facing the following Equations: $$\vec x_k = F\vec x_{k-1} + B\vec u_k$$ $$P_k = FP_{k-1}F^T + Q$$ $\vec x_k$: state vector. $P_k$: ...
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Confusion on allowed operation on row vectors

Assume we are given the following scalar quantity: \begin{align} q &= v^T A w \\ & =(v_1 \,\, v_2)\begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{pmatrix}\begin{pmatrix} w_1\\ ...
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If I know rank-$1$ matrix ${\bf x}{\bf x}^\top$, how can I find $\| {\bf x} \|_2^2$?

If $A$ is a $3 \times 1$ matrix and $$AA^{T} = \begin{pmatrix} 1 & -1 & 1 \\ -1 & 1 & -1 \\ 1 & -1 & 1 \\ \end{pmatrix}$$ what is $...
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Do I transpose a matrix so I can multiply it? [closed]

So I have two matrices: [2x3] x [2x5] At their current state, these two matrices are unable to be multiplied. Right? Would I transpose the [2x3] matrix into a [3x2] so I can multiply it with the ...
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Is the the transformation of the transposed matrix equal to transpose of the transformed matrix?

Let $A$ be a square matrix expressed in a basis $\{\mathbf e_i\}$ and let $C$ be the transition matrix from $\{\mathbf e_i\}$ to a new basis $ \{\tilde{\mathbf e}_i\}$, so $\widetilde {A}=DAC$, with $...
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Is it possible for $AA^T$ to be a nilpotent matrix if neither $A$ nor $A^T$ are?

If $A$ is a square, non-nilpotent matrix with real-valued elements, and its transpose is $A^T$, then is it ever possible for $AA^T$ to be nilpotent? What if we allow complex-valued elements? Is $AA^H$ ...
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Symmetric matrices, transpose matrices [duplicate]

i have a problem from my linear algebra exam that i couldn't solve: $B$ is an element of $M_n(\mathbb{R})$ and we now that $B\cdot B=B$ and that $B^T \cdot B=B \cdot B^T$, we now have to show that ...
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linear algebra - linear transformation and transpose

Every linear transformation can be represented as a matrix. If I take a matrix representation of a certain linear transformation and transpose it, what kind of operator do I get? when does the ...
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Common factor in sum of transpose matrices

I have the following linear matrix inequality: $(B K_1)^T + BK_1 < -2A$ where B is 2x1, $K_1$ is 1x2 and A is 2x2. Is it possible to find $K_1$ as a common factor in the left-hand side of the ...
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Please tell me a simple proof for the following equality: $\operatorname{rank} A=\operatorname{rank} A A^T.$

Please tell me a simple proof for the following equality: $$\operatorname{rank} A=\operatorname{rank} A A^T.$$ I proved the above equality as follows: I found the following formula in "Linear ...
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Why do we define Symplectic groups with Transpose and not hermitian adjoint?

In the definition of a symplectic group over a field, we take the definition: all matrices $S$ of some dimension $2n$, such that $$S^T \Omega S = \Omega$$ where omega is a skew-symmetric matrix (bi-...
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What is the most common notation for a transpose of a gradient of a function?

Say we have a function where the argument is pretty large horizontally, like $f(x^k,\lambda_k, \mu_k)$, or something. Then you take the transpose of this gradient for whatever reason, be it for a ...
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Hat matrix: various results in linear algebra.

In the question, the hat matrix $H = I-x(x^Tx)^{-1}x^T$ is an $n$ x $n$ matrix. $x$ is a $n$ x $p$ vector, whose columns are linearly dependent. How do I show that $Hx = 0$, $H^T = H$, and $H^2 = H$. ...
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Positive definiteness [duplicate]

Let $A$ be a positive definite matrix is it true that also $A+A^T$ is positive definite? If it is true, how to prove it? I try to employ the definition using eigenvalues but i'm not able to proceed. ...
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What happens to the dependance of the columns of a matrix once manipulated?

Lets say that we have a square matrix ( n x n ), and Ax=B has only one solution (where B ∈ 𝑅 ). We can tell from the fact that Ax=B has only one solution that the columns of A will be linearly ...
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Geometric sum of transpose matrices

Is there any mapping between the geometric sum of $A$ and the geometric sum of $A^T$? To be precise are there conditions under which $(I-A)^{-1}$ is approximately close to $(I-A^T)^{-1}$, other than ...
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Consider the following matrices. Calculate the results of the operations below, where the operations are allowed. [duplicate]

I have added a image of the question below, but to explain: suppose you have two matrices, one called C and another called B, how do you work out (CB)^T? is it a matter of multiplying C and B and then ...
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Gradients of matrix expression

I trying to find the gradients of the following matrix function $$F(X,A,B) = X^T(XA+B),$$ where $X,A,B$ are some matrices. $$\nabla_A F = X^TX \quad ?$$ $$\nabla_B F = X^T \quad ?$$ $$\nabla_X F = 2XA+...
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Visualizing what a transpose matrix $A^t$ does to a vector as compared to $A$

I am reading a proof that has a part that I am trying to visualise. In it this is a step in a long line of equations: $$(A\vec{u})^t(\vec{v})=\vec{u}^t(A^t\vec{v})$$ While I can see that it is true ...
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$T^*$ is surjective implies $T$ is bounded below. [closed]

Let $X,Y$ be banach spaces and $T: X \to Y$ a bounded linear map. I'm trying to show that if $T^*$ (adjoint of $T$) is surjective, then $T$ is bounded below, i.e. ${\exists} c>0$ s.t $||T(x)|| \ge ...
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If $P$ is an orthogonal projection, prove that $P^+ = P$.

If $P$ is an orthogonal projection, prove that $P^+ = P$. Where '$^+$' indicates the Moore-Penrose pseudo-inverse. What we know is the $P$ is symmetric and idempotent. That is $P = P^2 = P^T$. I am ...
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If we got a non-singular matrix $A$, how is the operation $Ã = (A^T)^{-1}$ called? And why is a matrix $A$ orthogonal if $A = Ã$?

This was just introduced in our Differential Geometry course, but no name was given to this. In the end, it's just the inverse transpose matrix, but does it have its "own" name? Also, I don'...
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$M:\mathbb{R}^{5\times5} T: M\to M$ denote the operator $T (A)=\frac12(A-A^T)$, find kernel,range,nullity,rank,real eigenvalues and eigenvectors?

My question is $M:\mathbb{R}^{5\times 5} T:M \to M$ denote the operator $$T (A)=\frac12(A-A^T)$$ where superscript $T$ is the matrix transpose. How can I find kernel,range,nullity,rank,reel ...
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Significance of $N(T^t)=R(T)^0$.

I know that if $T:V\to W$ is a linear transformation where $V,W$ are finite dimensional.Then we have $Ker(T^t)=Im(T)^0$.But how to geometrically interpret this thing.What does it mean and why this has ...
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How do I turn a vector into a specific matrix in MATLAB (example in post)

I'm currently wracking my brains on how to solve this specific situation: I have a (20x1) vector that looks like: $$ H' = \begin{matrix}1&2&3&4&5& \dots & 19 & 20 \end{...
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Is $\operatorname{rank}(A) = \operatorname{rank}( A^T )$ always true?

The answer key for my exam says that it's not always true, but I can't seem to come up with an explanation for this or find a matrix that makes this equation untrue. Thanks
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Why is the orthogonal complement of col(A) = ker(A*)?

col(A) = column space of matrix A ker(A*) = kernel (nullspace) of matrix A* (where A* is the conjugate transpose which is just AT for real matrices.) Why does orthogonal complement col(A) = ker(A*) ? ...
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Which change of basis matrix makes a companion matrix similar to its transpose?

I know that a companion matrix is similar to its transpose e.g by Smith normal form. When the characteristic polynomial splits, companion matrices become Jordan blocks. A Jordan block is similar to ...
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The inverse of a permutation matrix is its transpose

I do this exercise but I don't know if this okay. $$P^{-1}_{\sigma}=P^{T}_{\sigma}=P_{\sigma^{-1}}$$ Where $P_{\sigma}=[e_{\sigma(1)},e_{\sigma(2)},...,e_{\sigma(n)}]$ The first thing that I do is ...
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How do you prove the adjoint of a linear operator in any abstract inner product space is related to the conjugate transpose of its matrix?

If you define the inner product $\langle u,v\rangle=u^Tv$, then the fact that $\langle Au,v\rangle=\langle u,A^T\rangle$ where $A$ is a matrix defined over $\Bbb{R}$ is easily seen. Likewise, if you ...
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About vector transpose $uu^T$

I have some questions regarding vector Transpose. Let $u \in \Bbb R^n$, such that $||u||=1$ and let $A=uu^T$. So I understand that $A$ is a $\Bbb n ~{\Bbb x}~\Bbb n$ matrix. I want to find $\Bbb {nul}...
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Abstract tensor index notation for matrix transpose as (1,1) tensor?

In this answer to this question about the transpose of a (1,1) tensor, the answerer gives the following equation: $$(A^T)_j{}^i=A^i{}_j$$ as the transpose of the tensor $A$. My question is, what does ...
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With $(AB^{-1})^T$, what comes first, the multiplication or the transpose?

In part of the question we have $(AB^{-1})^T$. My first thought was that I multiply $A$ and $B^{-1}$, then apply the transpose. But according to theory $(AB)^T = (B)^T(A)^T,$ so this says I should get ...
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Transpose $D^tf$ of operator derivation $D\in L(V,V)$ applied to a function $f$ which is the integral of $p\in V$.

I have to determine the transpose $D^tf$ of linear transformation $D\in L(V,V)$, where $V=\mathcal{P}(\mathbb{R})$, for $a, b, \in \mathbb{R}$, $f\in V^{*}$ defined by: $$f(p)=\int_a^bp(x)dx.$$ Since ...
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Calculating transposes for elements of $\operatorname{End}(\mathbb{R}^{\infty})$.

Here, $\mathbb{R}^{\infty}$ is the vector space of infinite sequences where only finitely many elements are nonzero. I could easily calculate this for a finite-dimensional vector space, but now when ...
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