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Question 10: Determine if \(P = \left[ {\begin{array}{*{20}{c}}1&{.2}\\0&{.8}\end{array}} \right]\) is a regular stochastic matrix.

Short Answer

Expert verified

\(P\)is not a regular stochastic matrix.

Step by step solution

01

Define a regular stochastic matrix

The stochastic matrix isregular if some matrix power \({p^k}\) contains only positive entries.

02

 Step 2: Determine whether the matrix is regular stochastic

Compute the matrix \({P^2}\) as shown below:

\(\begin{aligned}{}{P^2} &= \left( {\begin{aligned}{{}}1&{.2}\\0&{.8}\end{aligned}} \right)\left( {\begin{aligned}{{}}1&{.2}\\0&{.8}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{1 + 0}&{0.2 + 0.16}\\{0 + 0}&{0 + 0.64}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&{.36}\\0&{.64}\end{aligned}} \right)\end{aligned}\)

\({P^k} = \left( {\begin{aligned}{{}}1&{1 - {8^k}}\\0&{{{.8}^k}}\end{aligned}} \right)\)haszero as its \(\left( {2,1} \right)\) entry for all \(k\).

Thus, \(P\) is not a regular stochastic matrix.

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Most popular questions from this chapter

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

Question 18: Suppose A is a \(4 \times 4\) matrix and B is a \(4 \times 2\) matrix, and let \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) represent a sequence of input vectors in \({\mathbb{R}^2}\).

  1. Set \({{\mathop{\rm x}\nolimits} _0} = 0\), compute \({{\mathop{\rm x}\nolimits} _1},...,{{\mathop{\rm x}\nolimits} _4}\) from equation (1), and write a formula for \({{\mathop{\rm x}\nolimits} _4}\) involving the controllability matrix \(M\) appearing in equation (2). (Note: The matrix \(M\) is constructed as a partitioned matrix. Its overall size here is \(4 \times 8\).)
  2. Suppose \(\left( {A,B} \right)\) is controllable and v is any vector in \({\mathbb{R}^4}\). Explain why there exists a control sequence \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) in \({\mathbb{R}^2}\) such that \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

Given \(T:V \to W\) as in Exercise 35, and given a subspace \(Z\) of \(W\), let \(U\) be the set of all \({\mathop{\rm x}\nolimits} \) in \(V\) such that \(T\left( {\mathop{\rm x}\nolimits} \right)\) is in \(Z\). Show that \(U\) is a subspace of \(V\).

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

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