You work at a company called 'PulseTech'. The startup is about to launch a new smart wristband that has a photoplethysmogram (PPG) sensor that monitors heart rhythms. This sensor captures a reading every 60 seconds, represented by beats per minute (bpm).Imagine you possess this heart rate information for N individuals, gathered over a span of one day, and you save it in a vector hi for i = 1, 2, 3, ..., N. (a) Let \( H = (h_1, h_2, \ldots, h_N) \). What is the dimensionality of this matrix?(b) Define a matrix \( S \) whose entries are:\[(S)_{i,j} = \left( \frac{h_i^T h_j}{\|h_i\| \|h_j\|} \right)\]You observe that this matrix seems to have a "blocky" structure where the bottom right quarter of the matrix has notably large values, close to 1. What might this indicate about the users? Use at most 3 sentences, appealing to linear algebraic concepts.

Answered: (a) Let H = (h₁, h2, ..., hy). What is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

You work at a company called 'PulseTech'. The startup is about to launch a new smart wristband that has a photoplethysmogram (PPG) sensor that monitors heart rhythms. This sensor captures a reading every 60 seconds, represented by beats per minute (bpm).

Imagine you possess this heart rate information for N individuals, gathered over a span of one day, and you save it in a vector hi for i = 1, 2, 3, ..., N.

(a) Let \( H = (h_1, h_2, \ldots, h_N) \). What is the dimensionality of this matrix?

(b) Define a matrix \( S \) whose entries are:

\[
(S)_{i,j} = \left( \frac{h_i^T h_j}{\|h_i\| \|h_j\|} \right)
\]

You observe that this matrix seems to have a "blocky" structure where the bottom right quarter of the matrix has notably large values, close to 1. What might this indicate about the users? Use at most 3 sentences, appealing to linear algebraic concepts.

Expert Solution

Step 1: Solution for (a)

(a) The matrix H has a dimensionality of N x 1440, where N is the number of individuals, and 1440 represents the number of minutes in a day (24 hours x 60 minutes).