Post History
#2: Post edited
by
Josh Hyatt
·
2022-02-18T20:23:40Z (over 4 years ago)
Fix dimension of vector y
- 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
Josh Hyatt
·
2022-02-17T20:47:41Z (over 4 years ago)
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 $\vec{x}$?