Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Find the Cholesky factorization of the matrix

$A = [\begin{matrix} 4 & - 4 & 8 \\ - 4 & 13 & 1 \\ 8 & 1 & 26 \end{matrix}]$

Cholesky factorization

$A = [\begin{matrix} 2 & 0 & 0 \\ - 2 & 3 & 0 \\ 4 & 3 & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 4 \\ 0 & 3 & 3 \\ 0 & 0 & 1 \end{matrix}]$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information:

$R = [\begin{matrix} 4 & - 4 & 8 \\ - 4 & 13 & 1 \\ 8 & 1 & 26 \end{matrix}]$

Step 2: Determining Cholesky factorization of the matrix:

We've got

$|A^{(1)}| = 4 > 0, |A^{(2)}| = 4 (13) - 16 = 36 > 0 |A^{(3)}| = 4 (337) + 4 (- 112) + 8 (- 108) = 36 > 0$

All of the determinants of main submatrices are positive; hence A is positive definite and has a Cholesky factorization, according to theorem 8.2.5. Let

$L = [\begin{matrix} a & 0 & 0 \\ b & d & 0 \\ c & e & f \end{matrix}]$

be the lower triangular matrix with positive diagonal entries, i.e. a, d, f>0, with $A = {LL}^{T}$ . Thus,

${LL}^{T} = [\begin{matrix} a & 0 & 0 \\ b & d & 0 \\ c & e & f \end{matrix}] [\begin{matrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{matrix}] = [\begin{matrix} a^{2} & ab & ac \\ ba & b^{2} + d^{2} & bc + de \\ ca & cb + ed & c^{2} + e^{2} + f^{2} \end{matrix}] A = {LL}^{T} \Rightarrow [\begin{matrix} a^{2} & ab & ac \\ ba & b^{2} + d^{2} & bc + de \\ ca & cb + ed & c^{2} + e^{2} + f^{2} \end{matrix}] = [\begin{matrix} 4 & - 4 & 8 \\ - 4 & 13 & 1 \\ 8 & 1 & 26 \end{matrix}]$

By equating the various components, we arrive at

$a^{2} = 4 \Rightarrow a = 2 (> 0) ab = - 4 \Rightarrow b = - 2 ac = 8 \Rightarrow c = 4 b^{2} + d^{2} = 13 \Rightarrow d^{2} = 13 - 4 = 9 \Rightarrow d = 3 (> 0) bc + de = 1 \Rightarrow 3 e = 9 \Rightarrow e = 3 c^{2} + e^{2} + f^{2} = 26 \Rightarrow f^{2} = 26 - 16 = 9 = 1 \Rightarrow f = 1 (> 0)$

As a result, the Cholesky factorization of A is

$A = [\begin{matrix} 4 & - 4 & 8 \\ - 4 & 13 & 1 \\ 8 & 1 & 26 \end{matrix}] = [\begin{matrix} 2 & 0 & 0 \\ - 2 & 3 & 0 \\ 4 & 3 & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 4 \\ 0 & 3 & 3 \\ 0 & 0 & 1 \end{matrix}] = {LL}^{T}$

Step 3: Determining the Result:

Cholesky factorization

$A = [\begin{matrix} 2 & 0 & 0 \\ - 2 & 3 & 0 \\ 4 & 3 & 1 \end{matrix}] [\begin{matrix} 2 & - 2 & 4 \\ 0 & 3 & 3 \\ 0 & 0 & 1 \end{matrix}]$

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the Cholesky factorization of the matrix

$A = [\begin{matrix} 4 & - 4 & 8 \\ - 4 & 13 & 1 \\ 8 & 1 & 26 \end{matrix}]$

Step by Step Solution

Step 1: Given Information:

Step 2: Determining Cholesky factorization of the matrix:

Step 3: Determining the Result:

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the Cholesky factorization of the matrix A=[4-48-41318126]

Step by Step Solution

Step 1: Given Information:

Step 2: Determining Cholesky factorization of the matrix:

Step 3: Determining the Result:

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Find the Cholesky factorization of the matrix

$A = [\begin{matrix} 4 & - 4 & 8 \\ - 4 & 13 & 1 \\ 8 & 1 & 26 \end{matrix}]$