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Chapter 7: Q32E (page 395)

Suppose\(A = PR{P^{ - {\bf{1}}}}\), where P is orthogonal and R is upper triangular. Show that if A is symmetric, then R is symmetric and hence is actually a diagonal matrix.

Short Answer

Expert verified

Matrix R has to be a diagonal matrix.

Step by step solution

01

Simplify the equation \(A = PR{P^{ - {\bf{1}}}}\)

Multiply P on both sides of the equation \(A = PR{P^{ - 1}}\).

\(\begin{aligned}{}AP &= PR{P^{ - 1}}P\\AP &= PR\\{P^{ - 1}}AP &= {P^{ - 1}}PR\\{P^{ - 1}}AP &= R\\{P^T}AP &= R\end{aligned}\)

02

Take transpose on both sides of the equation \({P^T}AP = R\)

Take transpose on both sides of the equation \({P^T}AP = R\).

\(\begin{aligned}{}{\left( {{P^T}AP} \right)^T} &= {R^T}\\{P^T}{A^T}{\left( {{P^T}} \right)^T} &= {R^T}\\{P^T}{A^T}P &= {R^T}\\{P^T}AP &= {R^T}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{A^T} = A} \right)\\R &= {R^T}\end{aligned}\)

The above equation shows that R is symmetric and upper triangular.

Thus, R has to be a diagonal matrix.

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Most popular questions from this chapter

Question: [M] A Landsat image with three spectral components was made of Homestead Air Force Base in Florida (after the base was hit by Hurricane Andrew in 1992). The covariance matrix of the data is shown below. Find the first principal component of the data, and compute the percentage of the total variance that is contained in this component.

\[S = \left[ {\begin{array}{*{20}{c}}{164.12}&{32.73}&{81.04}\\{32.73}&{539.44}&{249.13}\\{81.04}&{246.13}&{189.11}\end{array}} \right]\]

In Exercises 17–24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) “diagonal” matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

24. Using the notation of Exercise 23, show that \({A^T}{u_j} = {\sigma _j}{v_j}\) for \({\bf{1}} \le {\bf{j}} \le {\bf{r}} = {\bf{rank}}\;{\bf{A}}\)

Question: In Exercises 1 and 2, convert the matrix of observations to mean deviation form, and construct the sample covariance matrix.

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Question: In Exercises 1 and 2, convert the matrix of observations to mean deviation form, and construct the sample covariance matrix.

\(1.\,\,\left( {\begin{array}{*{20}{c}}{19}&{22}&6&3&2&{20}\\{12}&6&9&{15}&{13}&5\end{array}} \right)\)

Question: 14. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Given any \({\rm{b}}\) in \({\mathbb{R}^m}\), adapt Exercise 13 to show that \({A^ + }{\rm{b}}\) is the least-squares solution of minimum length. [Hint: Consider the equation \(A{\rm{x}} = {\rm{b}}\), where \(\mathop {\rm{b}}\limits^\^ \) is the orthogonal projection of \({\rm{b}}\) onto Col \(A\).
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