Let A - = [5 2 0 -2 0-2 7 2 6 = PDPT where D = Compute (a) an orthogonal decomposition A decomposition of A, namely express A as the sum of rank 1 matrices A = \₁V₁V² + \2U₂U₁ + λ3Ú3V²¬ where P [V1 V2 V3] So that 7; are the normalized eigenvectors of A. Last, (c) write down QA(x) = ¹ Ar the quadratic form defined by A. 06 and (b) the spectral
Let A - = [5 2 0 -2 0-2 7 2 6 = PDPT where D = Compute (a) an orthogonal decomposition A decomposition of A, namely express A as the sum of rank 1 matrices A = \₁V₁V² + \2U₂U₁ + λ3Ú3V²¬ where P [V1 V2 V3] So that 7; are the normalized eigenvectors of A. Last, (c) write down QA(x) = ¹ Ar the quadratic form defined by A. 06 and (b) the spectral
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let
\[ A = \begin{bmatrix} 5 & 2 & 0 \\ 2 & 6 & -2 \\ 0 & -2 & 7 \end{bmatrix} \]
Compute (a) an orthogonal decomposition \( A = PDP^T \) where \( D = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 3 \end{bmatrix} \), and (b) the spectral decomposition of \( A \), namely express \( A \) as the sum of rank 1 matrices
\[ A = \lambda_1 \vec{v_1} \vec{v_1}^T + \lambda_2 \vec{v_2} \vec{v_2}^T + \lambda_3 \vec{v_3} \vec{v_3}^T \]
where \( P = [\vec{v_1} \ \vec{v_2} \ \vec{v_3}] \) so that \( \vec{v_i} \) are the normalized eigenvectors of \( A \). Last, (c) write down \( Q_A(\vec{x}) = \vec{x}^T A \vec{x} \) the quadratic form defined by \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b609ed5-0879-4e23-8eee-4515e84ef2b9%2F6d7097da-623e-4259-8a9b-6c3b10dd2ffb%2Fmw4dthf_processed.png&w=3840&q=75)
Transcribed Image Text:Let
\[ A = \begin{bmatrix} 5 & 2 & 0 \\ 2 & 6 & -2 \\ 0 & -2 & 7 \end{bmatrix} \]
Compute (a) an orthogonal decomposition \( A = PDP^T \) where \( D = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 3 \end{bmatrix} \), and (b) the spectral decomposition of \( A \), namely express \( A \) as the sum of rank 1 matrices
\[ A = \lambda_1 \vec{v_1} \vec{v_1}^T + \lambda_2 \vec{v_2} \vec{v_2}^T + \lambda_3 \vec{v_3} \vec{v_3}^T \]
where \( P = [\vec{v_1} \ \vec{v_2} \ \vec{v_3}] \) so that \( \vec{v_i} \) are the normalized eigenvectors of \( A \). Last, (c) write down \( Q_A(\vec{x}) = \vec{x}^T A \vec{x} \) the quadratic form defined by \( A \).
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