Let $T$ be an $m\times n$ matrix with column vectors $\vec{T_i}$:

$$ \vec{T_i}=\begin{bmatrix} \vec{T_1} & ... & \vec{T_n} \end{bmatrix} $$

Let $\vec{x}$ be an unknown $n$-element vector:

$$ \vec{x}=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} $$

Suppose the following equation holds for a known $m$-element vector $\vec{y}$:

$$ \frac{T\vec{x}}{\left|\left|T\vec{x}\right|\right|}= \frac{\vec{y}}{\left|\left|\vec{y}\right|\right|} $$

That is, $T\vec{x}$ and $\vec{y}$ have the same direction.

If each component $x_i$ of $\vec{x}$ must satisfy the condition $0\le x_i\le1$, how does one maximize the magnitude of $\vec{x}$?

posted over 2 years ago

CC BY-SA 4.0

2y ago

Josh Hyatt‭

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