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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.

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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What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices. For example,…
Zed
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Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix?
Void Star
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Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$

How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$? This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I am unsure how they are relevant.
jaynp
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Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting on socks and shoes is taking the shoes off, followed…
user1337
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Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero rows in the reduced row-echelon form of A". The…
Vivi
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Matrix is conjugate to its own transpose

Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose. I spent quite a bit of time on it now, and still could not prove it. At the risk of devaluing myself, might I ask someone else…
George
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A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I'm going back and forth between using the definitions…
coconutbandit
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Is every matrix conjugate to its transpose in a continuous way?

It is well-known that every square matrix is conjugate to its transpose. This means (in the case of real matrices) that, for each $n\times n$ matrix $M$ with real entries, there is a matrix $S_M\in GL(n,\mathbb{R})$ such that ${S_M}^{-1}MS_M=M^T$.…
José Carlos Santos
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Transpose of block matrix

I'm attempting to prove that $$ \left[ \begin{array}{c c} A & B \\ C & D \\ \end{array} \right]^\top = \left[ \begin{array}{c c} A^\top & C^\top \\ B^\top & D^\top \\ \end{array} \right]. $$ Intuitively, I can see that it's true. However, when I try…
Red
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What is the intuitive interpretation of the transpose compared to the inverse?

I've been thinking about this question already for a long time and I've just encountered it again in the following lemma: $$f(x) = g(Ax + b) \implies \nabla f = A^T \nabla g(Ax + b) $$ This lemma makes intuitive sense if you think of it as taking…
LionCoder
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How to prove $(AB)^T=B^T A^T$

Given an $m\times n$-matrix $A$ and an $n\times p$-matrix $B$. Prove that $(AB)^T = B^TA^T$. Here is my attempt: Write the matrices $A$ and $B$ as $A = [a_{ij}]$ and $B = [b_{ij}]$, meaning that their $\left(i,j\right)$-th entries are $a_{ij}$ and…
Jeff
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Product of a vector and its transpose (Projections)

I am doing a basic course on linear algebra, where the guy says $a^Ta$ is a number and $aa^T$ is a matrix not.m Why? Background: Say we are projecting a vector $b$ onto a vector $a$. By the condition of orthogonality, the dot product is…
Abhishek Bhatia
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Why multiply a matrix with its transpose?

This might be a very stupid question, but I do not seem to understand why I would multiple a matrix with its transpose. I am not a mathematician, but I am very interested in understanding the practical usage of equations: Imagine I have three…
Boro Dega
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Is matrix $A^TA$ always symmetric?

Through experience, I've seen that the following statement holds true: "$A^TA$ is always a symmetric matrix?", where $A$ is any matrix. However can this statement be proven/falsified?
paul23
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Why is the maximum Rayleigh quotient equal to the maximum eigenvalue?

(Note: I'm only interested in real-valued matrices here, so I'm using "transpose" and "symmetric" instead of the more general "transjugate" and "Hermitian" in the hope that it will simplify the proof. But the theorem apparently holds for…
Draconis
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