1) Suppose that A² = A and B is a diagonal matrix similar to A. Prove that B2 = B and that the entries on the diagonal of B are only 0 and 1.
1) Suppose that A² = A and B is a diagonal matrix similar to A. Prove that B2 = B and that the entries on the diagonal of B are only 0 and 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Problem Statement
1) **Matrix Similarity and Idempotence**
Suppose that \( A^2 = A \) and \( B \) is a diagonal matrix similar to \( A \). Prove that \( B^2 = B \) and that the entries on the diagonal of \( B \) are only 0 and 1.
### Explanation:
Let's break the problem into parts to understand the given conditions and the requirements:
1. \( A^2 = A \):
- This condition means that \( A \) is an idempotent matrix, implying \( A \) multiplied by itself results in \( A \).
2. \( B \) is a diagonal matrix similar to \( A \):
- Being similar to \( A \) means there exists an invertible matrix \( P \) such that \( B = P^{-1}AP \).
- Since \( B \) is diagonal, it simplifies the proof as we are dealing with its eigenvalues.
#### Proof Steps:
1. **Proving \( B^2 = B \)**:
- From the similarity transformation, \( B = P^{-1}AP \).
- To find \( B^2 \), consider:
\[
B^2 = (P^{-1}AP)(P^{-1}AP) = P^{-1}A(PP^{-1})AP = P^{-1}A^2P
\]
- Given \( A^2 = A \), substitute \( A \) for \( A^2 \):
\[
B^2 = P^{-1}AP = B
\]
- Therefore, \( B \) is also idempotent, i.e., \( B^2 = B \).
2. **Entries on the Diagonal of \( B \)**:
- Since \( B \) is a diagonal matrix, its entries are the eigenvalues of \( A \).
- From \( A^2 = A \), it follows that for each eigenvalue \(\lambda\) of \( A \):
\[
\lambda^2 = \lambda \implies \lambda (\lambda - 1) = 0
\]
- Thus, \(\lambda\) must be either 0 or 1.
- Hence, the entries on the diagonal of \( B \) must be only 0 or 1.
By following](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46a741e2-1f18-40f1-9eda-db0ce81998cf%2Fd121baa4-b42b-44cb-80b7-e8ec60473fdd%2Fcm21fa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
1) **Matrix Similarity and Idempotence**
Suppose that \( A^2 = A \) and \( B \) is a diagonal matrix similar to \( A \). Prove that \( B^2 = B \) and that the entries on the diagonal of \( B \) are only 0 and 1.
### Explanation:
Let's break the problem into parts to understand the given conditions and the requirements:
1. \( A^2 = A \):
- This condition means that \( A \) is an idempotent matrix, implying \( A \) multiplied by itself results in \( A \).
2. \( B \) is a diagonal matrix similar to \( A \):
- Being similar to \( A \) means there exists an invertible matrix \( P \) such that \( B = P^{-1}AP \).
- Since \( B \) is diagonal, it simplifies the proof as we are dealing with its eigenvalues.
#### Proof Steps:
1. **Proving \( B^2 = B \)**:
- From the similarity transformation, \( B = P^{-1}AP \).
- To find \( B^2 \), consider:
\[
B^2 = (P^{-1}AP)(P^{-1}AP) = P^{-1}A(PP^{-1})AP = P^{-1}A^2P
\]
- Given \( A^2 = A \), substitute \( A \) for \( A^2 \):
\[
B^2 = P^{-1}AP = B
\]
- Therefore, \( B \) is also idempotent, i.e., \( B^2 = B \).
2. **Entries on the Diagonal of \( B \)**:
- Since \( B \) is a diagonal matrix, its entries are the eigenvalues of \( A \).
- From \( A^2 = A \), it follows that for each eigenvalue \(\lambda\) of \( A \):
\[
\lambda^2 = \lambda \implies \lambda (\lambda - 1) = 0
\]
- Thus, \(\lambda\) must be either 0 or 1.
- Hence, the entries on the diagonal of \( B \) must be only 0 or 1.
By following
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
