Learn Before
Complex Vector Representation from Paired Real Vector Elements
A real-valued vector of dimension , represented by the sequence , can be reinterpreted as a complex-valued vector of dimension . This is accomplished by pairing adjacent elements of to form the real and imaginary parts of new complex numbers. The resulting vector is a sequence of these complex numbers, , where each element is defined as . The full transformation is expressed as:
This representation is a key step in applying rotational transformations to vectors, for example in Rotary Positional Embeddings (RoPE).
0
1
Tags
Ch.2 Generative Models - Foundations of Large Language Models
Foundations of Large Language Models
Foundations of Large Language Models Course
Computing Sciences
Related
Complex Vector Representation from Paired Real Vector Elements
Consider two sequences, A = (a₁, a₂, a₃, a₄) and B = (b₁, b₂, b₃, b₄). The elements are defined as a₁=9, a₂=3, a₃=7, a₄=3 and b₁=7, b₂=9, b₃=3, b₄=3. Which of the following statements accurately describes the relationship between sequence A and sequence B?
A data scientist is tasked with representing the daily closing price of a specific stock over a period of 10 consecutive trading days. Which of the following mathematical structures is the most appropriate for this task, and why?
Data Pairing from a Sequence
Learn After
Condensed Notation for Pairing Real Vector Elements into Complex Numbers
A real-valued vector of even dimension can be reinterpreted as a complex-valued vector of half the dimension by pairing adjacent elements. The first element of each pair forms the real part and the second element forms the imaginary part of a new complex number. For instance, the pair of real numbers
(a, b)would become the single complex numbera + ib. Given the real-valued vectorv = (8, -2, 0, 5, 1, 3), which of the following represents the correct transformation into a complex-valued vector?Real to Complex Vector Transformation
A transformation process creates a complex-valued vector by pairing adjacent elements of a real-valued vector of even dimension. The first element of each pair becomes the real part and the second becomes the imaginary part of a new complex number (e.g., the pair
(a, b)becomesa + ib). If this process results in the complex-valued vectorc = (4 + 2i, -1 + 5i, 0 - 8i), what was the original real-valued vector?