Let $A $ be an $n \times n $ matrix such that $AA^T \geq x^2I, x\geq 0 $, which means that the matrix $AA^T-x^2I$ is positive semidefinite. Can we show that $(A+A^T)/2 \geq xI$? Thanks

$$Q= \begin{bmatrix} \cos x & -\sin x\\ \sin x & \cos x\end{bmatrix}$$ Given x belongs to $\mathbb{R^2}$, show $Qx$ and $x$ are equidistant. I've tried dot producting $Qx$ and seeing whether they are equal. I just can't seem to get it.

Prove that if $A^t = A^{-1}$ then different rows of $A$ are orthogonal to each other I saw this statement in my linear algebra book and I don't really know how to prove it, I hope someone can help me out. Thank you in advance.

I'm trying to understand the cosine similarity in a simple and graphical way, very much similar to this question here but I do not understand how the person got to their answer.

We know that regular hexagons can tile the plane but not in a self-similar fashion. However we can construct a fractal known as a Gosper island, that has the same area as the hexagon but has the property that when surrounded by 6 identical copies produces a similar shape, but with dimensions scal...

Let be $$A=\begin{pmatrix} \frac{-3}{2} & 2 & \frac{-1}{2} \\ \frac{-1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & -2 & \frac{3}{2} \end{pmatrix}, B=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$$ Prove that A and B are similar. I know if we can find a matrix $P$ so that $A=P^{-1}...

I'm trying to calculate the vertex cosine similarity of a weighted directional graph, however struggling to understand the concept. While I understand the methodology for simple and directed graphs, weighted has me stumped. I'm following the examples provided by wolfram here however with the wei...

Consider two 3D polylines, A and B. I am interested in computing a distance/similarity between them (from their current positions, no need to find the "best overlap" first). I have come up with some reasonable approximations (average distance between all points on A to their closest points on B, ...

The given task is to find all 2x2 Matrices A that are similar to: a) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ b) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ c) $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ However, I don't really know which steps to t...

Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? Is there something that characterizes it in terms of cardinality? Is there a way to extend matri...

This seems like such a simple question, but I can't quite see a connection. I want to say that the two diagonal matrices can commute with each other, and this gives us the algebraic edge to show their similarity, but I'm generally bad with abstract algebra proofs.