2. Which of the following matrix can be covariance matrix and explain your results. 10 3 1 10 5 2 --- A = 2 50 B = 5 31 1 02 2 1 2 10 5 2 52 3 3 32
2. Which of the following matrix can be covariance matrix and explain your results. 10 3 1 10 5 2 --- A = 2 50 B = 5 31 1 02 2 1 2 10 5 2 52 3 3 32
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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![**Question 2:** Which of the following matrices can be a covariance matrix? Explain your results.
Matrices given:
Matrix \( A \):
\[
A = \begin{bmatrix}
10 & 3 & 1 \\
2 & 5 & 0 \\
1 & 0 & 2
\end{bmatrix}
\]
Matrix \( B \):
\[
B = \begin{bmatrix}
10 & 5 & 2 \\
5 & 3 & 1 \\
2 & 1 & 2
\end{bmatrix}
\]
Matrix \( C \):
\[
C = \begin{bmatrix}
10 & 5 & 2 \\
5 & 3 & 3 \\
2 & 3 & 2
\end{bmatrix}
\]
**Explanation:**
A covariance matrix must satisfy the following conditions:
- It is symmetric, meaning the element at row \( i \), column \( j \) must be equal to the element at row \( j \), column \( i \) for all \( i \) and \( j \).
- It is positive semidefinite, meaning for any non-zero vector \( x \), the product \( x^T A x \geq 0 \).
To determine which of these matrices could be a covariance matrix, you can check for symmetry and compute the eigenvalues to ensure they are non-negative for positive semidefiniteness.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab1a1587-d50d-4ba9-a864-2608a439d53e%2F29ad2023-4cb0-4469-b168-dc81b383e213%2Fhzp91ck_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 2:** Which of the following matrices can be a covariance matrix? Explain your results.
Matrices given:
Matrix \( A \):
\[
A = \begin{bmatrix}
10 & 3 & 1 \\
2 & 5 & 0 \\
1 & 0 & 2
\end{bmatrix}
\]
Matrix \( B \):
\[
B = \begin{bmatrix}
10 & 5 & 2 \\
5 & 3 & 1 \\
2 & 1 & 2
\end{bmatrix}
\]
Matrix \( C \):
\[
C = \begin{bmatrix}
10 & 5 & 2 \\
5 & 3 & 3 \\
2 & 3 & 2
\end{bmatrix}
\]
**Explanation:**
A covariance matrix must satisfy the following conditions:
- It is symmetric, meaning the element at row \( i \), column \( j \) must be equal to the element at row \( j \), column \( i \) for all \( i \) and \( j \).
- It is positive semidefinite, meaning for any non-zero vector \( x \), the product \( x^T A x \geq 0 \).
To determine which of these matrices could be a covariance matrix, you can check for symmetry and compute the eigenvalues to ensure they are non-negative for positive semidefiniteness.
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