Matrix Representation of a 2D Rotary Transformation
A 2D vector (\mathbf{x} = \begin{pmatrix} x_1 \ x_2 \end{pmatrix}) is transformed at position (i) with frequency (\theta) into a new vector (\mathbf{y} = \begin{pmatrix} y_1 \ y_2 \end{pmatrix}) according to the following equations:
(y_1 = x_1 \cos(i\theta) - x_2 \sin(i\theta)) (y_2 = x_1 \sin(i\theta) + x_2 \cos(i\theta))
Express this transformation as a matrix-vector multiplication of the form (\mathbf{y} = \mathbf{M} \mathbf{x}), where (\mathbf{M}) is a 2x2 matrix. Write out the matrix (\mathbf{M}).
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Ch.3 Prompting - Foundations of Large Language Models
Foundations of Large Language Models
Foundations of Large Language Models Course
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The application of a rotary positional embedding to a two-dimensional vector at position with frequency results in a new vector , where the components are calculated as:
Based on this structure, which statement best analyzes how the output vector's components are formed?
A positional encoding method transforms a two-dimensional vector at position into a new vector using the equations:
This transformation is considered non-linear with respect to the input vector because it involves trigonometric functions.
Matrix Representation of a 2D Rotary Transformation