Suppose A is the symmetric matrix below: A has the eigenvalues and unit vectors: The spectral decomposition of A has the form: Suppose: What is the value of p11? What is the value of 911? A = 2₁ = 9₁ v₁ = √√√2 2₂ = − 9₁ V₂ = √3/2 A = A₁ + A₂ = 2₁V₁ V² + 2₂ V₂V/ A A2 = 1₂V₂v² = = P11 P12 P21 P22/ 911 912 921 922
Suppose A is the symmetric matrix below: A has the eigenvalues and unit vectors: The spectral decomposition of A has the form: Suppose: What is the value of p11? What is the value of 911? A = 2₁ = 9₁ v₁ = √√√2 2₂ = − 9₁ V₂ = √3/2 A = A₁ + A₂ = 2₁V₁ V² + 2₂ V₂V/ A A2 = 1₂V₂v² = = P11 P12 P21 P22/ 911 912 921 922
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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![## Spectral Decomposition of a Symmetric Matrix
### Given Matrix A:
Suppose \( A \) is the symmetric matrix below:
\[
A = \begin{bmatrix} 0 & 9 \\ 9 & 0 \end{bmatrix}
\]
### Eigenvalues and Unit Vectors:
\( A \) has the eigenvalues and unit vectors:
\[
\lambda_1 = 9, \quad v_1 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]
\[
\lambda_2 = -9, \quad v_2 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -1 \end{bmatrix}
\]
### Spectral Decomposition of A:
The spectral decomposition of \( A \) has the form:
\[
A = A_1 + A_2 = \lambda_1 v_1 v_1^T + \lambda_2 v_2 v_2^T
\]
### Components of Decomposition:
Suppose:
\[
A_1 = \lambda_1 v_1 v_1^T = \begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix}
\]
\[
A_2 = \lambda_2 v_2 v_2^T = \begin{pmatrix} q_{11} & q_{12} \\ q_{21} & q_{22} \end{pmatrix}
\]
### Questions:
1. What is the value of \( p_{11} \)?
2. What is the value of \( q_{11} \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b609ed5-0879-4e23-8eee-4515e84ef2b9%2F5fa2134a-bcff-4cb9-b35b-cc38dd8a0cd1%2Ft3fqvvd_processed.png&w=3840&q=75)
Transcribed Image Text:## Spectral Decomposition of a Symmetric Matrix
### Given Matrix A:
Suppose \( A \) is the symmetric matrix below:
\[
A = \begin{bmatrix} 0 & 9 \\ 9 & 0 \end{bmatrix}
\]
### Eigenvalues and Unit Vectors:
\( A \) has the eigenvalues and unit vectors:
\[
\lambda_1 = 9, \quad v_1 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]
\[
\lambda_2 = -9, \quad v_2 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -1 \end{bmatrix}
\]
### Spectral Decomposition of A:
The spectral decomposition of \( A \) has the form:
\[
A = A_1 + A_2 = \lambda_1 v_1 v_1^T + \lambda_2 v_2 v_2^T
\]
### Components of Decomposition:
Suppose:
\[
A_1 = \lambda_1 v_1 v_1^T = \begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix}
\]
\[
A_2 = \lambda_2 v_2 v_2^T = \begin{pmatrix} q_{11} & q_{12} \\ q_{21} & q_{22} \end{pmatrix}
\]
### Questions:
1. What is the value of \( p_{11} \)?
2. What is the value of \( q_{11} \)?
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