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Q&A Maximize Independent Variable of Matrix Multiplication

1 answer · posted 2y ago by Josh Hyatt‭ · edited 2y ago by Josh Hyatt‭

Question matrices vector

#2: Post edited by $user avatar$ Josh Hyatt‭ · 2022-02-18T20:23:40Z (over 2 years ago)
Fix dimension of vector y

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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 $n$-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 $\vec{x}$?

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 $\vec{x}$?

#1: Initial revision by $user avatar$ Josh Hyatt‭ · 2022-02-17T20:47:41Z (over 2 years ago)

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Maximize Independent Variable of Matrix Multiplication

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 $n$-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 $\vec{x}$?

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