# 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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### Proof about Hadamard matrices that $H H^\textsf{T} = n I_n$ [closed]

I have read the Wikipedia article about Hadamard matrices that says: Let $H$ be a Hadamard matrix of order $n$, the following is true: $H H^\textsf{T} = n I_n$, where $I_n$ is the identity $n×n$ ...
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### Product of Conjugate Transpose with Itself equalling the Zero Matrix. [duplicate]

I am reading a linear algebra book and I am stuck on one of the questions that asks to (a) : Find all nxn matrices with real entries such that $$A^{T}A = 0$$ (b) : Find all nxn matrices with complex ...
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### Find a matrix $Q$ that has $Q(Q^T)=I$ but does not have orthonormal columns

My textbook says that if a matrix $Q$ is square and has orthonormal columns then $Q(Q^T)=I$, but it does not say the opposite (that if $Q(Q^T)=I$ then $Q$ has orthonormal columns). Is there an example ...
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### Prove $\| A\|_2=\|A^T\|_2$

Let $A \in \mathbb{R}^{n \times n},$ Prove $\| A\|_2=\|A^T\|_2$ Similarly as the 1-norm I could show that $\| A\|_2=\max_{i} \sqrt{\sum |a_{ij}|^2}$. But this result does not help me to prove this ...
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### Why a matrix has to be invertible for $\operatorname{adj} A^T=(\operatorname{adj}A)^T$ to be true?

I read a theorem If $A$ is an invertible square matrix, then $\operatorname{adj} A^T= (\operatorname{adj} A)^T$. But after attempting to prove it myself and also reading the proof I am unable to ...
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### Dot product between two vectors transformed by orthogonal matrices

I am reading through the "Matrix Transformations" chapter of this book and more specifically on Orthogonal Matrices. I understand their properties and understand that multiplying vectors by ...
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### When is $A^TA$ invertible?

Let $A$ be a matrix. What are some necessary/sufficient conditions for the Gram matrix $A^T A$ to be invertible? This question came up when I was trying to learn about least-squares regression. Is it ...
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### For a square matrix $A$ over any field, does $A^TA=0$ imply $AA^T=0$?

If $A^TA=cI$ for some scalar $c\neq0$, then $A^T=cA^{-1}$, and thus $AA^T=cI$ as well. (The first equation says that $c^{-1}A^T$ is a left-inverse of $A$. And a left-inverse is also a right-inverse: ...
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### Matrix made up by $2$ orthonormal vectors $x$ and $y$ in $\mathbb{R^2}$ and $y$ is transposed

I came across the following exercise from my professor and had no idea how to start. I tried it with the standard basis in $\mathbb{R}^{2}$, which results in a rank of $1$ and eigenvalue $0$, but this ...
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### Is $\mathbf{A}\mathbf{A}^T$ symmetric? [duplicate]

Is matrix $$\mathbf{B}=\mathbf{A}\mathbf{A}^T$$ necessarily symmetric (i.e., does $\mathbf{B}=\mathbf{B}^T$)? Writing $$b_{ij}=a_{ij}a_{ji}$$ seems wrong because the $j$ on the LHS is different from ...
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### Intuition behind thinking of the transpose of matrices?

Not directly related to the title, but I recently went through LADR, and did not find it particularly intuitive. I've started to work through LADW. I've found it much more intuitive. There is one ...
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### Solve for vector $r$ in an equation involving sum of the square elements of $r$ and the sum of the square elements of $Mr$ where $M$ is a known matrix

Hello stackexchange community. I have a system of equations where the aim is finding two vectors $r$ and $c$ of lengths $t$ and $m$ respectively. There is a known matrix $M$ of shape $m\times t$ and ...
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### Conceptualizing and Contextualizing Quadratic Forms

In an 'Estimation Theory' course that I'm taking, I notice over and over that the equations we work with are littered with what I think are called 'Quadratic Forms'. In case that terminology is ...
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### Is congruency same as $*$-congruency for real matrices?
Two real matrices $A,B\in M_n(\mathbb R)$ are said to be congruent to each other if there exists an invertible matrix $P\in GL_n(\mathbb R)$ such that $B=P^tAP$, where $P^t$ is the transpose of $P$. ...
### Knowing solution of ${\bf A x} = {\bf b}$, find solution for ${\bf A}^\top {\bf x} = {\bf b}$ [closed]
Suppose that ${\bf A} \in \Bbb R^{n \times n}$ is invertible, ${\bf b} \in \Bbb R^{n}$. Knowing a solution of ${\bf A} {\bf x}_1 = {\bf b}$, find a solution for ${\bf A}^\top {\bf x}_2 = {\bf b}$. ...