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Chapter 5: Q5.3-29E (page 267)

Question: A factorization \(A = PD{P^{ - {\bf{1}}}}\) is not unique. Demonstrate this for the matrix A in Example 2. With \({D_{\bf{1}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{0}}\\{\bf{0}}&{\bf{5}}\end{array}} \right]\), use the information in Example 2 to find the matrix \({P_{\bf{1}}}\) such that \(A = {P_{\bf{1}}}{D_{\bf{1}}}P_{\bf{1}}^{ - {\bf{1}}}\).

Short Answer

Expert verified

The matrix \({P_1}\) is \(\left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\).

Step by step solution

01

Write the information given

The matrices from example 2 are,

\[A = \left[ {\begin{array}{*{20}{c}}7&2\\{ - 4}&1\end{array}} \right], P = \left[ {\begin{array}{*{20}{c}}1&1\\{ - 1}&{ - 2}\end{array}} \right], D = \left[ {\begin{array}{*{20}{c}}5&0\\0&3\end{array}} \right]\]

02

Find the matrix \({P_{\bf{1}}}\)

For eigenvalue 5, the eigenvector is \(\left[ {\begin{array}{*{20}{c}}1\\{ - 1}\end{array}} \right]\) and for eigenvalue 3, the eigenvector is \(\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right]\).

The matrix \({P_1}\) is such that \(A = {P_1}{D_1}P_1^{ - 1}\), so the matrix \({D_1}\) is:

\[{D_1} = \left[ {\begin{array}{*{20}{c}}3&0\\0&5\end{array}} \right]\]

Interchange the columns of P to find the matrix \({P_1}\).

\[{P_1} = \left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\]

03

Find the product \({P_{\bf{1}}}{D_{\bf{1}}}P_{\bf{1}}^{ - {\bf{1}}}\)

The product \({P_1}{D_1}P_1^{ - 1}\) can be calculated as follows:

\[\begin{array}{c}{P_1}{D_1}P_1^{ - 1} = \left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3&0\\0&5\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 1}\\2&1\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}7&2\\{ - 4}&1\end{array}} \right]\\ = A\end{array}\]

So, the matrix \({P_1}\) is \(\left[ {\begin{array}{*{20}{c}}1&1\\{ - 2}&{ - 1}\end{array}} \right]\).

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Most popular questions from this chapter

Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

6. \(\left[ {\begin{array}{*{20}{c}}3&- 4\\4&8\end{array}} \right]\)

[M]Repeat Exercise 25 for \[A{\bf{ = }}\left[ {\begin{array}{*{20}{c}}{{\bf{ - 8}}}&{\bf{5}}&{{\bf{ - 2}}}&{\bf{0}}\\{{\bf{ - 5}}}&{\bf{2}}&{\bf{1}}&{{\bf{ - 2}}}\\{{\bf{10}}}&{{\bf{ - 8}}}&{\bf{6}}&{{\bf{ - 3}}}\\{\bf{3}}&{{\bf{ - 2}}}&{\bf{1}}&{\bf{0}}\end{array}} \right]\].

19–23 concern the polynomial \(p\left( t \right) = {a_{\bf{0}}} + {a_{\bf{1}}}t + ... + {a_{n - {\bf{1}}}}{t^{n - {\bf{1}}}} + {t^n}\) and \(n \times n\) matrix \({C_p}\) called the companion matrix of \(p\): \({C_p} = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{1}}&{\bf{0}}&{...}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{}&{\bf{0}}\\:&{}&{}&{}&:\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{}&{\bf{1}}\\{ - {a_{\bf{0}}}}&{ - {a_{\bf{1}}}}&{ - {a_{\bf{2}}}}&{...}&{ - {a_{n - {\bf{1}}}}}\end{aligned}} \right)\).

23. Let \(p\) be the polynomial in Exercise \({\bf{22}}\), and suppose the equation \(p\left( t \right) = {\bf{0}}\) has distinct roots \({\lambda _{\bf{1}}},{\lambda _{\bf{2}}},{\lambda _{\bf{3}}}\). Let \(V\) be the Vandermonde matrix

\(V{\bf{ = }}\left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{1}}&{\bf{1}}\\{{\lambda _{\bf{1}}}}&{{\lambda _{\bf{2}}}}&{{\lambda _{\bf{3}}}}\\{\lambda _{\bf{1}}^{\bf{2}}}&{\lambda _{\bf{2}}^{\bf{2}}}&{\lambda _{\bf{3}}^{\bf{2}}}\end{aligned}} \right)\)

(The transpose of \(V\) was considered in Supplementary Exercise \({\bf{11}}\) in Chapter \({\bf{2}}\).) Use Exercise \({\bf{22}}\) and a theorem from this chapter to deduce that \(V\) is invertible (but do not compute \({V^{{\bf{ - 1}}}}\)). Then explain why \({V^{{\bf{ - 1}}}}{C_p}V\) is a diagonal matrix.

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

  1. If \(A\) is \(3 \times 3\), with columns \({{\rm{a}}_1}\), \({{\rm{a}}_2}\), and \({{\rm{a}}_3}\), then \(\det A\) equals the volume of the parallelepiped determined by \({{\rm{a}}_1}\), \({{\rm{a}}_2}\), and \({{\rm{a}}_3}\).
  2. \(\det {A^T} = \left( { - 1} \right)\det A\).
  3. The multiplicity of a root \(r\) of the characteristic equation of \(A\) is called the algebraic multiplicity of \(r\) as an eigenvalue of \(A\).
  4. A row replacement operation on \(A\) does not change the eigenvalues.

Mark each statement as True or False. Justify each answer.

a. If \(A\) is invertible and 1 is an eigenvalue for \(A\), then \(1\) is also an eigenvalue of \({A^{ - 1}}\)

b. If \(A\) is row equivalent to the identity matrix \(I\), then \(A\) is diagonalizable.

c. If \(A\) contains a row or column of zeros, then 0 is an eigenvalue of \(A\)

d. Each eigenvalue of \(A\) is also an eigenvalue of \({A^2}\).

e. Each eigenvector of \(A\) is also an eigenvector of \({A^2}\)

f. Each eigenvector of an invertible matrix \(A\) is also an eigenvector of \({A^{ - 1}}\)

g. Eigenvalues must be nonzero scalars.

h. Eigenvectors must be nonzero vectors.

i. Two eigenvectors corresponding to the same eigenvalue are always linearly dependent.

j. Similar matrices always have exactly the same eigenvalues.

k. Similar matrices always have exactly the same eigenvectors.

I. The sum of two eigenvectors of a matrix \(A\) is also an eigenvector of \(A\).

m. The eigenvalues of an upper triangular matrix \(A\) are exactly the nonzero entries on the diagonal of \(A\).

n. The matrices \(A\) and \({A^T}\) have the same eigenvalues, counting multiplicities.

o. If a \(5 \times 5\) matrix \(A\) has fewer than 5 distinct eigenvalues, then \(A\) is not diagonalizable.

p. There exists a \(2 \times 2\) matrix that has no eigenvectors in \({A^2}\)

q. If \(A\) is diagonalizable, then the columns of \(A\) are linearly independent.

r. A nonzero vector cannot correspond to two different eigenvalues of \(A\).

s. A (square) matrix \(A\) is invertible if and only if there is a coordinate system in which the transformation \({\bf{x}} \mapsto A{\bf{x}}\) is represented by a diagonal matrix.

t. If each vector \({{\bf{e}}_j}\) in the standard basis for \({A^n}\) is an eigenvector of \(A\), then \(A\) is a diagonal matrix.

u. If \(A\) is similar to a diagonalizable matrix \(B\), then \(A\) is also diagonalizable.

v. If \(A\) and \(B\) are invertible \(n \times n\) matrices, then \(AB\)is similar to \ (BA\ )

w. An \(n \times n\) matrix with \(n\) linearly independent eigenvectors is invertible.

x. If \(A\) is an \(n \times n\) diagonalizable matrix, then each vector in \({A^n}\) can be written as a linear combination of eigenvectors of \(A\).
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