/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Let \(S\) be the \(1 \times n\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 19

Let \(S\) be the \(1 \times n\) row matrix with a 1 in each column, \(S=\left[\begin{array}{llll}{1} & {1} & {\cdots} & {1}\end{array}\right]\) a. Explain why a vector \(\mathbf{x}\) in \(\mathbb{R}^{n}\) is a probability vector if and only if its entries are nonnegative and \(S \mathbf{x}=1 .\) (A \(1 \times 1\) matrix such as the product \(S \mathbf{x}\) is usually written without the matrix bracket symbols.) b. Let \(P\) be an \(n \times n\) stochastic matrix. Explain why \(S P=S .\) c. Let \(P\) be an \(n \times n\) stochastic matrix, and let \(\mathbf{x}\) be a probability vector. Show that \(P \mathbf{x}\) is also a probability vector.

Short Answer

Expert verified
a) Sx=1 sums to 1; non-negativity for probability. b) SP=S due to column sums. c) Px is a probability vector due to preservation.

Step by step solution

01

Understand Probability Vectors

A probability vector \( \mathbf{x} \in \mathbb{R}^{n} \) must satisfy two conditions: each entry \( x_i \geq 0 \) (non-negativity), and the sum of all entries equals 1. If \( S = [1, 1, \cdots, 1] \), the product \( S\mathbf{x} \) equals \( \sum_{i=1}^{n} x_i \). Hence, \( S\mathbf{x} = 1 \) ensures that the sum of all entries in \( \mathbf{x} \) is 1.
02

Define Stochastic Matrix

An \( n \times n \) stochastic matrix \( P \) has columns that are probability vectors, meaning each column sums up to 1. Thus, for each column of the matrix \( P \), multiplying it by \( S \) results in the sum of its elements being 1, therefore \( S P = S \).
03

Prove Probability Preservation

To show \( P \mathbf{x} \) is a probability vector, we need to show \( S(P\mathbf{x}) = 1 \) and that all entries are non-negative.1. Since \( S \mathbf{x} = 1 \), we have \( S(P \mathbf{x}) = S(P) \mathbf{x} = S \mathbf{x} = 1 \), thanks to Step 2.2. Each element of \( P\mathbf{x} \) is a linear combination of the non-negative entries of \( \mathbf{x} \) with non-negative coefficients from \( P \), thus ensuring non-negativity. Hence, \( P\mathbf{x} \) is a probability vector.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stochastic Matrices
Imagine a grid where each column has numbers that add up to 1. That's what defines a stochastic matrix. Simply put, each column is a probability vector. This means, regardless of how big the matrix is, each vertical slice should total to 1.
Stochastic matrices are crucial in fields like Markov chains where they model state transitions.
  • Each column behaves like a probability vector
  • Ensures total transition probability equals 1
  • Useful in predicting probabilities over time, like weather forecasts or stock prices
Understanding this framework helps you grasp how systems evolve and probabilities are conserved across transitions.
Non-Negativity
In the world of probability and matrices, non-negativity is a comforting friend. It means that all numbers you work with are zero or positive. Think of it like a safety net ensuring probabilities never dip below zero, which wouldn't make sense.
Non-negativity is a condition that a probability vector must satisfy. When you have a vector with all non-negative entries, each part actually represents a possible outcome in a set, all sandwiched between zero and one.
  • Keeps probabilities valid, preventing negative outcomes
  • Ensures logical consistency in probability calculations
Working with positive entries ensures your calculations align with real-world phenomena.
Matrix Multiplication
Matrix multiplication might sound complicated, but it's quite fascinating! It's like combining two patterns to create a new one. Here's how it works in our context: you've got a vector (a list of numbers) and a matrix (a grid of numbers). You multiply them together to mix their information.
In terms of probabilities, multiplying a matrix by a vector can transform that vector. In our case, using a stochastic matrix transforms a probability vector to another probability vector, thanks to its unique properties.
  • The columns and rows interact to create a new vector
  • Maintains the integrity of probabilities through transformation
Through this process, you can model how initial states change over time.
Vector Spaces
Picture vector spaces as an entire universe of possibilities in terms of directions and magnitudes. Each vector in this space can be added or scaled to explore new realms within it. A space containing probability vectors means that any movement within this space will still respect the primary rules: non-negativity and sum to one.
Vector spaces give us a structured way to study combinations of vectors. They provide a playground where vectors like our probability vector can be worked on through operations like addition and scalar multiplication, all while remaining within defined boundaries.
  • Structured arena for vector operations
  • Allows for manipulation while preserving fundamental properties
This concept is key in understanding how different arrangements and combinations of probabilities behave under various conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Exercises \(27-29\) concern an \(m \times n\) matrix \(A\) and what are often called the fundamental subspaces determined by \(A .\) Which of the subspaces Row \(A,\) Col \(A,\) Nul \(A, \operatorname{Row} A^{T}\) , Col \(A^{T},\) and \(\mathrm{Nul} A^{T}\) are in \(\mathbb{R}^{m}\) and which are in \(\mathbb{R}^{n} ?\) How many distinct subspaces are in this list?

At time \(k=0,\) an initial investment of \(\$ 1000\) is made into a savings account that pays 6\(\%\) interest per year compounded monthly. (The interest rate per month is \(005 . )\) Each month after the initial investment, an additional \(\$ 200\) is added to the account. For \(k=0,1,2, \ldots,\) let \(y_{k}\) be the amount in the account at time \(k,\) just after a deposit has been made. a. Write a difference equation satisfied by \(\left\\{y_{k}\right\\}\) b. \([\mathbf{M}]\) Create a table showing \(k\) and the total amount in the savings account at month \(k,\) for \(k=0\) through \(60 .\) List your program or the keystrokes you used to create the table. c. \([\mathbf{M}]\) How much will be in the account after two years (that is, 24 months), four years, and five years? How much of the five-year total is interest?

Is the following difference equation of order 3\(?\) Explain. \(y_{k+3}+5 y_{k+2}+6 y_{k+1}=0\)

Exercises 17 and 18 concern a simple model of the national economy described by the difference equation $$ Y_{k+2}-a(1+b) Y_{k+1}+a b Y_{k}=1 $$ Here \(Y_{k}\) is the total national income during year \(k, a\) is a constant less than \(1,\) called the marginal propensity to consume, and \(b\) is a positive constant of adjustment that describes how changes in consumer spending affect the annual rate of private investment. Find the general solution of equation \((14)\) when \(a=.9\) and \(b=.5 .\)

In Exercises \(5-8,\) find the steady-state vector. $$ \left[\begin{array}{ccc}{.7} & {.2} & {.2} \\ {0} & {.2} & {.4} \\ {.3} & {.6} & {.4}\end{array}\right] $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.