In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.

Is

Concepcion Hale
2021-12-15
Answered

Is a matrix multiplied with its transpose something special?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.

Is$\mathrm{\forall}}^{T$ something special for any matrix A?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.

Is

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Let $a=(a1,a2,a3)$ be a fixed vector in ${\mathbb{R}}^{3}$. Define the cross product $a\times v$ of $a$ and another vector $v=({v}_{1},{v}_{2},{v}_{3})\in {\mathbb{R}}^{3}$ as

$a\times v=det\left[\begin{array}{ccc}{e}_{1}& {e}_{2}& {e}_{3}\\ {a}_{1}& {a}_{2}& {a}_{3}\\ {v}_{1}& {v}_{2}& {v}_{3}\end{array}\right]$

Define a function $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ by $T(v)=a\times v$ for $v\in {\mathbb{R}}^{3}$.

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$a\times v=det\left[\begin{array}{ccc}{e}_{1}& {e}_{2}& {e}_{3}\\ {a}_{1}& {a}_{2}& {a}_{3}\\ {v}_{1}& {v}_{2}& {v}_{3}\end{array}\right]$

Define a function $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ by $T(v)=a\times v$ for $v\in {\mathbb{R}}^{3}$.

a) Show that $T$ is a matrix transformation and calculate its representing matrix $M$

b) Find $\mathrm{ker}(T)$ and interpret its answer geometrically

c) find $\mathrm{range}(T)$ and interpret its answer geometrically

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Let $X$ be a $m\times n$ (m: number of records, and n: number of attributes) normalized dataset (between 0 and 1). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand if $R$ was drawn randomly from Gaussian distribution, e.g.,$N(0,1)$ then the transformation preserve the Euclidean distances between instances (all of the pairwise distances between the points in the feature space will be preserved). But what if $R\sim U(0,1)$, does the transformation still preserve the distance between instances?