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Chapter 7: Q26E (page 395)

Show that if an \(n \times n\) matrix \(A\) is positive definite, then there exists a positive definite matrix \(B\) such that \(A = {B^T}B\). (Hint: Write \(A = PD{P^T}\), with\({P^T} = {P^{ - 1}}\). Produce a diagonal matrix \(C\) such that \(D = {C^T}C\), and let \(B = PC{P^T}\). Show that \(B\) works.)

Short Answer

Expert verified

It is proved that if a \(n \times n\) matrix \(A\) is positive definite, then there exists a positive definite matrix \(B\) such that \({B^T}B = A\).

Step by step solution

01

Symmetric Matrices and Quadratic Forms

When any Symmetric Matrix \(A\) is diagonalized orthogonallyas\(PD{P^{ - 1}}\) we have:

\(\begin{aligned}{}{x^T}Ax = {y^T}Dy\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{\rm{as }}x = Py} \right\}\\{\rm{and}}\\\left\| x \right\| = \left\| {Py} \right\| = \left\| y \right\|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\;\;\;\left\{ {\forall y \in \mathbb{R}} \right\}\end{aligned}\)

02

Show that if a \(n \times n\) matrix \(A\) is positive definite, then there exists a positive definite matrix \(B\) such that \({B^T}B = A\)

As per the question, we have:

The positive\(n \times n\)matrix\(A\),Let we have:

\(A = PD{P^T},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{\rm{where }}{P^T} = {P^{ - 1}}} \right\}\)

For\(C\)being its diagonal matrix with all positive eigenvalues, we have:

\(\begin{aligned}{}D &= {C^2}\\ &= {C^T}C\end{aligned}\)

For,\(B = PC{P^T}\), we have:

\(\begin{aligned}{}{B^T}B &= {\left( {PC{P^T}} \right)^T}\left( {PC{P^T}} \right)\\ &= \left( {{P^T}{C^T}{P^T}^T} \right)\left( {PC{P^T}} \right)\\ &= P{C^T}C{P^T}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{P^T}P = 1} \right\}\\ &= PD{P^T}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{C^T}C = D} \right\}\\ &= A\end{aligned}\)

Henceit is proved thatif a \(n \times n\) matrix \(A\) is positive definite, then there exists a positive definite matrix \(B\) such that \({B^T}B = A\).

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Most popular questions from this chapter

Question 8: Use Exercise 7 to show that if A is positive definite, then A has a LU factorization, \(A = LU\), where U has positive pivots on its diagonal. (The converse is true, too).

(M) Orhtogonally diagonalize the matrices in Exercises 37-40. To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and for each eigenvalue \(\lambda \), find an orthogonal basis for \({\bf{Nul}}\left( {A - \lambda I} \right)\), as in Examples 2 and 3.

39. \(\left( {\begin{aligned}{{}}{.{\bf{31}}}&{.{\bf{58}}}&{.{\bf{08}}}&{.{\bf{44}}}\\{.{\bf{58}}}&{ - .{\bf{56}}}&{.{\bf{44}}}&{ - .{\bf{58}}}\\{.{\bf{08}}}&{.{\bf{44}}}&{.{\bf{19}}}&{ - .{\bf{08}}}\\{ - .{\bf{44}}}&{ - .{\bf{58}}}&{ - .{\bf{08}}}&{.{\bf{31}}}\end{aligned}} \right)\)

Compute the quadratic form \({x^T}Ax\), when \(A = \left( {\begin{aligned}{{}}3&2&0\\2&2&1\\0&1&0\end{aligned}} \right)\) and

a. \(x = \left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right)\)

b. \(x = \left( {\begin{aligned}{{}}{ - 2}\\{ - 1}\\5\end{aligned}} \right)\)

c. \(x = \left( {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right)\)

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17鈥�22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

16. \(\left( {\begin{aligned}{{}}{\,6}&{ - 2}\\{ - 2}&{\,\,\,9}\end{aligned}} \right)\)

In Exercises 17鈥�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 \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

18. Suppose \(A\) is square and invertible. Find a singular value decomposition of \({A^{ - 1}}\)
See all solutions

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