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Chapter 4: Q16SE (page 191)

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.)

Short Answer

Expert verified

It is proved that \({\mathop{\rm rank}\nolimits} \left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

Step by step solution

01

Show that \({\mathop{\rm rank}\nolimits} \left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\)

Consider rank\(A = {r_1}\) and rank\(B = {r_2}\). Rank factorization of \(A\) and \(B\) are \(A = {C_1}{R_1}\) and \(B = {C_2}{R_2}\), where \({C_1}\) is \(m \times {r_1}\) with rank\({r_1}\), \({C_2}\) is \(m \times {r_2}\) with rank\({r_2}\), \({R_1}\) is \({r_1} \times n\) with rank\({r_1}\), \({R_2}\) is \({r_2} \times n\) with rank\({r_2}\).

By stacking \({R_1}\) over \({R_2}\), create a \(m \times \left( {{r_1} + {r_2}} \right)\) matrix \(C = \left( {\begin{array}{*{20}{c}}{{C_1}}&{{C_2}}\end{array}} \right)\) and a \(\left( {{r_1} + {r_2}} \right) \times n\) matrix \(R\).

\(\begin{array}{c}A + B = {C_1}{R_1} + {C_2}{R_2}\\ = \left( {\begin{array}{*{20}{c}}{{C_1}}&{{C_2}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{R_1}}\\{{R_2}}\end{array}} \right)\\ = CR\end{array}\)

The matrix \(CR\) is a product, so according to exercise 12, its rank cannot exceed the rank of either of its factors. The rank of \(C\) cannot exceed \({r_1} + {r_2}\) because \(C\) has \({r_1} + {r_2}\). Similarly, as matrix \(R\) has \({r_1} + {r_2}\) rows, then its rank cannot exceed \({r_1} + {r_2}\).

Therefore, the rank of \(A + B\) cannot exceed \({r_1} + {r_2} = {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\), or \({\mathop{\rm rank}\nolimits} \left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

Thus, it is proved that \({\mathop{\rm rank}\nolimits} \left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

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Most popular questions from this chapter

Suppose a \({\bf{5}} \times {\bf{6}}\) matrix A has four pivot columns. What is dim Nul A? Is \({\bf{Col}}\,A = {\mathbb{R}^{\bf{3}}}\)? Why or why not?

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Let be a basis of\({\mathbb{R}^n}\). .Produce a description of an \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)matrix A that implements the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\). Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)). (See Exercise 21.)

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

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)

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 \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).
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