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.

In linear algebra, the transpose of a matrix A is another matrix B created by any of the below equivalent actions:

  • Reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain the transpose.
  • Write the rows of A as the columns of the transpose.
  • Write the columns of A as the rows of the transpose.

This tag is to be used for questions related to the transpose operation, specifically an inquiry into its properties or special characteristics of it.

532 questions
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Is the space of maps which satisfy this vanishing condition finite-dimensional?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Let $h:\mathbb{D}^n \to \mathbb{R}^{k}$ be smooth, and suppose that $h(x) \neq 0$ a.e. on $\mathbb{D}^n$. Set $$V_h=\{ \,\,f \in…
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Derivative of transpose

I am trying to find derivative of this : RQ(u) = uTXTXu / uTu I need help finding derivative : RQ/u Optimal sol should satisfy XTXu = RQ(u)u I am very confused, any help would be great. If you could share some references that will be very helpful…
Ella
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Is there any relation between an eigenvector of $A$ and the eigenvector of $A^T$ with the same eigenvalue?

Let $A$ be a square matrix over $\mathbb C$, and let $A^T$ denote its transpose. It is not hard to see that $A$ and $A^T$ have the same set of eigenvalues, so given $Ax=\lambda x$ for some vector $x\in V$ and eigenvalue $\lambda\in\mathbb C$, we…
glS
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$|\text{det}(A)| = 1$ implies $A$ is orthogonal

I know that $A$ orthogonal $\Rightarrow$ |det($A$)| = 1. Now I need to prove or disprove the reversed statement: $$ |\det(A)| = 1 \Rightarrow A \,\text{ is orthogonal} $$ This is what I'm currently trying: $$ |\det(A)| = 1 \Rightarrow \det(A)^2 = 1…
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Can the same vector be an eigenvector of both $A$ and $A^T$?

It is proven that $A$ and $A^T$ have the same eigenvalues. I want to study what stands for eigenvectors. Let me make a try. Given: $$Ax=\lambda x$$ we know that $x\in C(A)$ for $\lambda \neq 0$. Suppose that for $A^T$ we have the same eigenvectors…
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Prove $x^TAx = 0$ $\implies$ A is skew-symmetric

We know for a skew-symmetric matrix A, $x^TAx = 0$. But is the converse statement true, i.e. does $x^TAx = 0$ imply A is skew-symmetric? If yes, then how to prove it?
SM10
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Geometric intepretation of transpose of matrix

Whenever we see a matrix $A=\bigl( \begin{smallmatrix} 3 & 2 \\ 1 & 2 \end{smallmatrix} \bigr)$ and $v=(3, 2),$ we can visualize that $(3, 2)$ represent the coordinates of $\mathbf i$ vector and $(1,2)$ represents the $\mathbf j$ vector. And to…
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Can an Perron eigenvector of a non-symmetric irreducible nonnegative matrix be also a Perron eigenvector of its transpose?

Let $\mathbf{X}$ be nonnegative irreducible matrix such that $\mathbf{X} \ne \mathbf{X}^T$. Let $\mathbf{p}(\mathbf{A})$ denote a right eigenvector corresponding to the Perron root of $\mathbf{A}$, $\rho(\mathbf{A})$. Similarly,…
owovrokfop
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Difference between Adjoint of a matrix and its transpose

For simplicity let $T:R^{n}\rightarrow R^{n},$ be a linear operator and $[A]_T$ be the matrix of operator $T.$ Then matrix of adjoint $T^{\times}$ operator of $T$ is given as $$[A]_{T^{\times}}=[A]_{T}^t$$ where $t$ denotes transpose of a…
Taj
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When should I take conjugate transpose of a complex matrix, and when transpose of it?

I was taking the inverse of $$A=\begin{bmatrix} 2+i &1 \\ 1&-2+i \end{bmatrix}$$ and $\det(A)=-6 $, and cofactor matrix $$C=\begin{bmatrix} -2+i &-1 \\ -1&2+i \end{bmatrix}$$ such that correct way to do it is…
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eigenvalues of $AA^T$ and $A^TA$

Is it true (and under which conditions) that the products of an non-square matrix $A$ and its transpose and vice versa (so the product of the transpose and $A$) share the same eigenvalues (multiplicities omitted)?
Zoran
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Show that arbitrary $A$ and $A^T$ have same eigenvalue, algebraic and geometric multiplicity

Show that an arbitrary $n \times n$ matrix $A$ and its transpose $A^T$ have the same eigenvalues, algebraic multiplicity and geometric multiplicity. I'm not sure if I did it correctly and especially how to show that they have same geometric…
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When can we say that $A^{\mathrm T} B = B^{\mathrm T} A$?

I was looking at the derivation of the normal equation from here. Now, the author has used the fact that $A^{\mathrm T} B = B^{\mathrm T} A$ to reach the step shown in the below image. Can anyone provide some information like, when is it true, or…
Sourajit
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Vectorization and transpose: how are $\text{vec}(W^T)$ and $\text{vec}(W)$ related?

In solving for a gradient, I ended up with a differential that looks similar to: $$ dT = (a^T \otimes b^T)\ \text{vec}[d[W]^T] + (b^T \otimes c^T)\ \text{vec}[d[W]] $$ and I am trying to solve for $\frac{\partial T}{\partial \text{vec}W}$. The…
Robert
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Let A be an $m\times n$ matrix. Prove that $\operatorname{rank}(AA^T) = \operatorname{rank}(A)$.

Let $A$ be an $m\times n$ matrix. Prove that $\operatorname{rank}(AA^T) = \operatorname{rank}(A)$. The problem tells me to prove it with the theorem that $\operatorname{rank}(A^TA) = \operatorname{rank}(A)$. I'm a bit lost here...$AA^T$ and $(A)$…
Tommy Ling
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