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Chapter 2: Q19SE (page 93)

(M) Certain dynamical systems can be studied by examining powers of a matrix, such as those below. Determine what happens to \({A^k}\) and \({B^k}\) as \(k\) increases (for example, try \(k = 2,...,16\)). Try to identify what is special about \(A\) and \(B\). Investigate large powers of other matrices of this type, and make a conjecture about such matrices.

\(A = \left( {\begin{aligned}{*{20}{c}}{.4}&{.2}&{.3}\\{.3}&{.6}&{.3}\\{.3}&{.2}&{.4}\end{aligned}} \right),B = \left( {\begin{aligned

Short Answer

Expert verified

As \(k\) increases, the matrices \({{\mathop{\rm A}\nolimits} ^k}\) and \({{\mathop{\rm B}\nolimits} ^k}\) become \({{\mathop{\rm A}\nolimits} ^k} = \left( {\begin{aligned}{*{20}{c}}{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\\{\frac{3}{7}}&{\frac{3}{7}}&{\frac{3}{7}}\\{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\end{aligned}} \right)\), \({{\mathop{\rm B}\nolimits} ^k} = \left( {\begin{aligned}{*{20}{c}}{\frac{{18}}{{89}}}&{\frac{{18}}{{89}}}&{\frac{{18}}{{89}}}\\{\frac{{33}}{{89}}}&{\frac{{33}}{{89}}}&{\frac{{33}}{{89}}}\\{\frac{{38}}{{89}}}&{\frac{{38}}{{89}}}&{\frac{{38}}{{89}}}\end{aligned}} \right)\).

Step by step solution

01

Determine what happens to \({A^k}\) as k increases

Consider the matrix\(A = \left( {\begin{aligned}{*{20}{c}}{.4}&{.2}&{.3}\\{.3}&{.6}&{.3}\\{.3}&{.2}&{.4}\end{aligned}} \right)\).

Use the MATLAB code to compute the matrices \({A^2},{A^4},{A^8}\) as shown below:

\(\begin{aligned}{l} > > {\mathop{\rm A}\nolimits} = \left( {.4\,\,\,.2\,\,\,.3;\,\,.3\,\,\,.6\,\,\,.3;\,\,.3\,\,\,.2\,\,\,.4} \right)\\ > > {\mathop{\rm A}\nolimits} \^2\end{aligned}\)

\({{\mathop{\rm A}\nolimits} ^2} = \left( {\begin{aligned}{*{20}{c}}{.31}&{.26}&{.30}\\{.39}&{.48}&{.39}\\{.30}&{.26}&{.31}\end{aligned}} \right)\)

\(\begin{aligned}{l} > > {\mathop{\rm A}\nolimits} \^2 = \left( {.31\,\,\,.26\,\,\,.30;\,\,.39\,\,\,.48\,\,\,.39;\,\,.30\,\,\,.26\,\,\,.31} \right)\\ > > {\mathop{\rm A}\nolimits} \^4 = {\mathop{\rm A}\nolimits} \^2 * {\mathop{\rm A}\nolimits} \^2\end{aligned}\)

\({{\mathop{\rm A}\nolimits} ^4} = \left( {\begin{aligned}{*{20}{c}}{.2875}&{.2834}&{.2874}\\{.4251}&{.4332}&{.4251}\\{.2874}&{.2834}&{.2875}\end{aligned}} \right)\)

\(\begin{aligned}{l} > > {\mathop{\rm A}\nolimits} \^4 = \left( {.2857\,\,.2857\,\,\,.2857;\,\,.4285\,\,.4286\,\,\,.4285;\,\,.2857\,\,\,.2857\,\,\,.2857} \right)\\ > > {\mathop{\rm A}\nolimits} \^8 = {\mathop{\rm A}\nolimits} \^4 * {\mathop{\rm A}\nolimits} \^4\end{aligned}\)

\({{\mathop{\rm A}\nolimits} ^8} = \left( {\begin{aligned}{*{20}{c}}{.2857}&{.2857}&{.2857}\\{.4285}&{.4286}&{.4285}\\{.2857}&{.2857}&{.2857}\end{aligned}} \right)\)

As \(k\)increases, the four decimal places become

\({{\mathop{\rm A}\nolimits} ^k} \to \left( {\begin{aligned}{*{20}{c}}{.2857}&{.2857}&{.2857}\\{.4286}&{.4286}&{.4286}\\{.2857}&{.2857}&{.2857}\end{aligned}} \right)\).

The rational format of the matrix \({A^k}\) is shown below:

\({{\mathop{\rm A}\nolimits} ^k} \to \left( {\begin{aligned}{*{20}{c}}{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\\{\frac{3}{7}}&{\frac{3}{7}}&{\frac{3}{7}}\\{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\end{aligned}} \right)\)

02

Determine what happens to \({B^k}\) as k increases

Consider the matrix.

Use the MATLAB code to compute the matrices as shown below:

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Most popular questions from this chapter

Let \(X\) be \(m \times n\) data matrix such that \({X^T}X\) is invertible., and let \(M = {I_m} - X{\left( {{X^T}X} \right)^{ - {\bf{1}}}}{X^T}\). Add a column \({x_{\bf{0}}}\) to the data and form

\(W = \left[ {\begin{array}{*{20}{c}}X&{{x_{\bf{0}}}}\end{array}} \right]\)

Compute \({W^T}W\). The \(\left( {{\bf{1}},{\bf{1}}} \right)\) entry is \({X^T}X\). Show that the Schur complement (Exercise 15) of \({X^T}X\) can be written in the form \({\bf{x}}_{\bf{0}}^TM{{\bf{x}}_{\bf{0}}}\). It can be shown that the quantity \({\left( {{\bf{x}}_{\bf{0}}^TM{{\bf{x}}_{\bf{0}}}} \right)^{ - {\bf{1}}}}\) is the \(\left( {{\bf{2}},{\bf{2}}} \right)\)-entry in \({\left( {{W^T}W} \right)^{ - {\bf{1}}}}\). This entry has a useful statistical interpretation, under appropriate hypotheses.

In the study of engineering control of physical systems, a standard set of differential equations is transformed by Laplace transforms into the following system of linear equations:

\(\left[ {\begin{array}{*{20}{c}}{A - s{I_n}}&B\\C&{{I_m}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\bf{x}}\\{\bf{u}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{y}}\end{array}} \right]\)

Where \(A\) is \(n \times n\), \(B\) is \(n \times m\), \(C\) is \(m \times n\), and \(s\) is a variable. The vector \({\bf{u}}\) in \({\mathbb{R}^m}\) is the “input” to the system, \({\bf{y}}\) in \({\mathbb{R}^m}\) is the “output” and \({\bf{x}}\) in \({\mathbb{R}^n}\) is the “state” vector. (Actually, the vectors \({\bf{x}}\), \({\bf{u}}\) and \({\bf{v}}\) are functions of \(s\), but we suppress this fact because it does not affect the algebraic calculations in Exercises 19 and 20.)

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Use partitioned matrices to prove by induction that for \(n = 2,3,...\), the \(n \times n\) matrices \(A\) shown below is invertible and \(B\) is its inverse.

\[A = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\1&1&0&{}&0\\1&1&1&{}&0\\ \vdots &{}&{}& \ddots &{}\\1&1&1& \ldots &1\end{array}} \right]\]

\[B = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\{ - 1}&1&0&{}&0\\0&{ - 1}&1&{}&0\\ \vdots &{}& \ddots & \ddots &{}\\0&{}& \ldots &{ - 1}&1\end{array}} \right]\]

For the induction step, assume A and Bare \(\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrices, and partition Aand B in a form similar to that displayed in Exercises 23.

Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{3}}&{ - {\bf{4}}}\\{\bf{7}}&{ - {\bf{8}}}\end{aligned}} \right)\).

Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).
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