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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Truly intuitive geometric interpretation for the transpose of a square matrix

I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how $A^{T}$ relates to A. I want to be able to instantly visualize in my mind what I'm doing to the space when…
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How do you transpose tensors?

We transpose a matrix $A$ by replacing $A_{ij}$ with $A_{ji}$, for all $i$ and $j$. However, in case $A$ has more than two dimensions (that is, it is a tensor), I don't know how to apply the transpose operation. If A has dimensions $3\times 3…
Nirvana
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Help with proving that the transpose of the product of any number of matrices is equal to the product of their transposes in reverse

Specifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I'm stuck: To prove the theorem I would like to show that ((An)T)ij = ((AT)n)ij for all ij. All I can think of is…
1west
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If $A$ is a matrix such that $A^T = A^2$, what are eigenvalues of $A$?

If $A$ is a matrix such that $A^T = A^2$, what are eigenvalues of $A$? Now I read somewhere that changing the matrix by taking a transpose does not change the characteristic polynomial. So it is safe to say that the annihilating polynomial in this…
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For every *non-square* matrix prove that $AA^t$ or/and $A^tA$ is singular

For every non-square matrix prove that $AA^t$ or/and $A^tA$ is singular. Like the title, I want to prove this and I tried to think of ways to prove it but I couldn't think of some.. I know by this answer that $AA^t$ is symmetric but I cant make the…
Zik332
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Minimal polynomial of $T(A) = A^\top - A$

As said in the title , I need to find the minimal polynomial of the linear transformation $$T(A)=A^\top-A.$$ The matrices are $M_n(\mathbb{C})$. I've figured out that $T^2 = 2A - 2A^t$ , so a polynomial $p(t) = t^2 + 2t$ works so $p(T) = 0$. Now…
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Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same eigenvectors. I have seen around some posts, but…
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Adding a diagonal matrix to a product of transpose of a matrix and itself is always invertible

I am asking this question in context to Regularization/Ridge Regression Let's say that there is a Matrix A of dimension n x d, where n is the number of rows and d is the number of columns ( n may or may not be larger than d) Consequently, we cannot…
cph_sto
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Prove that $\det(A+B)\det(A-B)=0$

Let $A,B$ be two $n\times n$ matrices with real entries, where $n$ is odd, such that $A\cdot A^{t}=I_n$ and $B\cdot B^{t}=I_n$. Prove that $$\det(A+B)\det(A-B)=0$$ It is obvious that $A^{-1}=A^{t}$ and $B^{-1}=B^{t}$, so $\det A, \det B = \pm 1$.…
Shroud
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Can you transpose a matrix using matrix multiplication?

Say you have a matrix A = \begin{bmatrix}a&b\\c&d\end{bmatrix} and I want it to look like $A^T$ = \begin{bmatrix}c&a\\d&b\end{bmatrix} Can this be done via matrix multiplication? Something like a matrix T such that $T*A = A^T$.
yujinred
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Does negative transpose sign mean inverse of a transposed matrix or transpose of an inverse matrix?

I want to know meaning of $$H^{-T}$$Is it same with $$(H^{-1})^T$$or $$(H^T)^{-1}$$
Salim Azak
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For $ A \in M_{m \times n}(\mathbb{R})$, does $\ker(A)$ relate to $\ker(A^T) $?

For $ A \in M_{m \times n}(\mathbb{R})$, does $\ker(A)$ relate to $\ker(A^T) $? And if so, what would be the connection? I wouldn't imagine there being any without additional conditions on $A$ (like symmetry or something), but I'm not sure. To…
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Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A}) = \sum\limits_{\pi \in {S_n}} {{\mathop{\rm sgn}} (\pi ) \cdot {a_{\pi (1),1}}} ...{a_{\pi…
AnnieOK
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Proving:$\operatorname{Proj}_{U^\perp}(x)=-\frac1{\det(A^TA)} X(u_1,\ldots, u_{n-2}, X(u_1,\ldots, u_{n-2}, x))$

The problem I'm trying to solve is as follows, which was posed to me by my professor as an exercise: Let $x, u_i \in \Bbb R^n$, $ A = (u_1, u_2, \ldots, u_{n-2})$ and $\{u_1, u_2, \ldots, u_{n-2}\}$ is linearly independent. Let $U = \text{Col}(A)$.…
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Prove that matrix $A\in \mathbb{R}^{n \times n}$ is invertible if $A^T = p(A)$

I have to prove that a matrix $A\in \mathbb{R}^{n \times n}$ is invertible if $A^T = p(A)$ where $p(A)$ is a polynomial with non-zero last coefficient. I've tried to use that if $A^T=p(A)$ then $A=p(A^T)$ and to look at $$AA^T =…
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