Let g: R -> R and v, w in R^n such that g(v_i) = w_i for i in [n]. Find an expression for g such that diag(v)^-1 = diag(w).
Understand the Problem
The question is asking to find a function g that maps elements of vector v to elements of vector w such that when the reciprocal of the diagonal matrix formed by v is computed, it equals the diagonal matrix formed by w.
Answer
$$ g(v_i) = \frac{1}{v_i} $$
Answer for screen readers
The function $g$ that maps elements of vector $v$ to elements of vector $w$ is given by:
$$ g(v_i) = \frac{1}{v_i} $$
Steps to Solve
- Define the vectors and diagonal matrices
Let's assume vector $v$ is given by $v = [v_1, v_2, \ldots, v_n]$ and vector $w$ is given by $w = [w_1, w_2, \ldots, w_n]$.
The diagonal matrices can be formed as:
$$ D_v = \begin{pmatrix} v_1 & 0 & \cdots & 0 \ 0 & v_2 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & v_n \end{pmatrix} $$
$$ D_w = \begin{pmatrix} w_1 & 0 & \cdots & 0 \ 0 & w_2 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & w_n \end{pmatrix} $$
- Find the reciprocal of the diagonal matrix formed by v
To find the reciprocal of the diagonal matrix $D_v$, we take the reciprocal of each diagonal element:
$$ D_v^{-1} = \begin{pmatrix} \frac{1}{v_1} & 0 & \cdots & 0 \ 0 & \frac{1}{v_2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & \frac{1}{v_n} \end{pmatrix} $$
- Set the equation to find g(v)
To satisfy the condition that the reciprocal of $D_v$ equals $D_w$, we have the equation:
$$ D_v^{-1} = D_w $$
This implies that for each $i$, we have:
$$ \frac{1}{v_i} = w_i $$
- Solve for g(v)
To relate vectors $v$ and $w$, we can express $w_i$ in terms of $v_i$:
$$ w_i = \frac{1}{v_i} $$
Thus, we can define the function $g$ that maps $v$ to $w$:
$$ g(v_i) = \frac{1}{v_i} $$
The function $g$ that maps elements of vector $v$ to elements of vector $w$ is given by:
$$ g(v_i) = \frac{1}{v_i} $$
More Information
This solution reveals a key relationship between the reciprocal of the elements in vector $v$ and the corresponding elements of vector $w$. Using such mapping functions is common in linear algebra, particularly with matrices and transformations.
Tips
- Confusing the operation of taking the reciprocal with other matrix operations.
- Forgetting to apply the reciprocal to each element individually in the vector.
To avoid these mistakes, double-check each step to ensure the operation is performed on each individual element correctly.
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