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.

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Skew-symmetric matrices

Prove that if $A$ is an $n\times n$ matrix, then $A - A^T$ is a skew-symmetric matrix. Thank you!
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What is the difference between transpose and inverse?

I have 2 tasks: To show that $A^{-1} = A^T$ and that $A^T = A^{-1}$. So I proved the first case with: $$A^T A = I$$ and later according to uniqueness of inverse for matrices we can say that they are equal. But what about the second case $A^T =…
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Gilbert Strang, Linear Algebra, Problem 27 Section 3.3

Problem set 3 from MIT 18.06 in Spring 2010 (solutions on OCW) includes the following exercise (which is problem 27 in Section 3.3 of Gilbert Strang, Introduction to Linear Algebra, 4th ed. 2009): Suppose $R$ is $m$ by $n$ of rank $r$, with pivot…
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If $x$ and $y\in\Bbb{R}^{n}$ are eigenvectors for $\lambda\neq\mu$, respectively, show $x^{T}\cdotp y = 0$

For $x^{T}\cdotp y = 0$, I understand that I can either look at it through matrix multiplication $x^{T}y^{T} = 0$ as you can't do that multiplication. I'm very sure this isn't the right way of looking at it but am unsure how else to think about…
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True or False questions about invertible matrices

If $A$ is invertible, then $(A^T)^{-1}= (A^{-1})^T $ If $A$ and $B$ are invertible, then $A+B$ is also invertible and its inverse is $A^{-1} +B^{-1}$ Note: Given a matrix $A$, the inverse and the transpose of $A$ are denoted $A^{-1}$ and $A^T$…
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Generalized Inverse of Transpose

How can we show that if $G$ is generalized Inverse of $A$ , then $G^T$ is generalized Inverse of $A^T$
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Consider the following matrices. Calculate the results of the operations below, where the operations are allowed.

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 transposing or is it a matter of B transposed…
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