Articles of transpose

interpreting the effect of transpose in the normal equations

I have a question about the normal equation. $A$ an $m\times n$ matrix with trivial nullspace, $y$ a vector outside the range of $A$. The vector $x$ that minimizes $|| Ax – y ||^2$ is the solution to $A^*Az = A^*y$. How can I interpret what $A^*$ is doing? Why does multiplying $Ax = y$ […]

Let $A$ be an 8 x 5 matrix of rank 3, and let $b$ be a nonzero vector in $N(A^T)$. Show $Ax=b$ must be inconsistent.

Here’s the entire question: Let $A$ be an 8 x 5 matrix of rank 3, and let $b$ be a nonzero vector in $N(A^T)$. a) Show that the system $Ax = b$ must be inconsistent. Gonna take a wild stab at this one… If the rank is 3, that means the dimension of the column […]

what does it mean for the transpose of a matrix to be the negative of the matrix?

Say I have matrix A,if $A^{T} = A$ then A is symmetrical. But what does it mean if $A^{T} = -A$ and can I know something about such a matrix inverse?

Prove $A^tB^t = (BA)^t$

Just going over some old homework problems for my test tomorrow. One of the questions was the prove $A^tB^t = (BA)^t$ and at the time I was really unsure of my answer and wrote the following: My Answer: We can write the $ij^{th}$ entry of $(BA)$ as $(BA)_{ij} = \sum_{k=1}^m{b_{ik}a_{kj}}$. Coincidentally, $(A^tB^t)_{ij}=\sum_{k=1}^n(A^t)_{ik}(B^t)_{kj} = \sum_{k=1}^na_{ki}b_{jk}$ Is […]

To create a special matrix !!

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$. Where $N$ is a scalar and $\mathbb{I}_N$ is the $N \times N$ identity matrix. Examle for 2×2 matrix $-1$ $1$ $1$ $1$ Example […]

$A^TA$ is always a symmetric matrix?

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/falsefied?

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, if I have a linear transformation, say on the plane, my intuition is […]

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 lecturer then explained that if the matrix […]

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 of rank: $\operatorname{rank} (A) = \dim(\operatorname{col}(A)) = \dim(\operatorname{row}(A))$ or using the rank theorem that […]

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?