Articles of matrices

Singular Value Decomposition of Rank 1 matrix

I am trying to understand singular value decomposition. I get the general definition and how to solve for the singular values of form the SVD of a given matrix however, I came across the following problem and realized that I did not fully understand how SVD works: Let $0\ne u\in \mathbb{R}^{m}$. Determine an SVD for […]

Each eigenvalue of $A$ is equal to $\pm 1$. Why is $A$ similar to $A^{-1}$?

This question already has an answer here: If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ – 1}}$ 1 answer

Can every $T$-stable subspace be realised as the kernel of another linear operator that commutes with$~T$?

This is inspired by the question Is Every Invariant Subspace the Kernel of an polynomial applied in the operator?, where “invariant subspace” and “polynomial in” are relative to a given linear operator$~T$ on a finite dimensional vectors space. The answer to that question is a simple “no”, because of simple examples like scalar operators, which […]

Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn’t include the full proofs. That’s fine for the question as it stands, but I find the subject interesting enough to wonder about the details. So here I’m asking. Let’s start by defining […]

Matrix Theory book Recommendations

I’m currently reading Sheldon Axler’s “Linear Algebra Done Right”. Can anyone recommend any good books on matrix theory at about the same level that might compliment it?

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the Roll and it is \begin{pmatrix} cos(Pitch) sin(Yaw)\\ cos(Yaw) cos(Pitch)\\ sin(Pitch) \end{pmatrix}. How should it […]

Minimize $\|AXBd -c \|^2$, enforcing $X$ to be a diagonal block matrix

Currently, I am minimizing the quadratic objective $\|\mathbf{A}\mathbf{X}\mathbf{B}\mathbf{d} -\mathbf{c} \|^2$ using CVX, as follows echo on cvx_begin variable xx(num_triangles*3,num_triangles*3) minimize( norm( A * xx * B * d – c ) ) cvx_end echo off However, $\mathbf{X}$ is a very large matrix (about $50,000 \times 50,000$, which is too big). Good news is that $\mathbf{X}$ […]

How to determine all vectors $b$ for which $Ax = b$ has a solution? Do the columns of $A$ span $\mathbb{R}^3$?

Let $A = \left(\begin{array}{rrr|r} 1&1&-15&36\\ 1&2&-10&41\\ 1&2&-9&42 \end{array}\right)$. Here is the row reduction: $A = \left(\begin{array}{rrr|r} 1&0&0&51\\ 0&1&0&0\\ 0&0&1&1 \end{array}\right)$. a) Determine all vectors $b$ for which $Ax = b$ has a solution. Write your answer as the span of a set of linearly independent vectors. (I have no idea what this is asking) b) […]

Upper Triangular Form of a Matrix

I am trying to find the upper triangular form of $B$ and an invertible matrix $C$ such that $B=C^{-1}AC$ where A is given by the following: $A = \pmatrix{1&1\\ -1&3}$ The characteristic equation is $(x-2)^2$ with eigenvalue $2$ and eigenvector $\langle1, 1\rangle$. Finding an upper triangular matrix similar to $A$ means to find a vector […]

Finding $A,B\in SL_2(\Bbb{Z})$ of finite order with the property that $AB=C$ where the order of $C$ is infinite.

I’m trying to think of two matrices $A,B\in SL_2(\Bbb{Z})$ of finite order ($A^n=B^m=I$) with the property that $AB=C$ where the order of $C$ is infinite. I guess that just by trial and error I could find two of those matrices, but I would like to find those in a little bit more sophisticated way. What […]