Prove that if A is idempotent, i.e. that A? = A, then either A = I or A is singular. %3D
Prove that if A is idempotent, i.e. that A? = A, then either A = I or A is singular. %3D
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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![**Proof of Idempotent Matrix Properties**
**Statement:**
Prove that if \( A \) is idempotent, i.e., that \( A^2 = A \), then either \( A = I \) or \( A \) is singular.
**Explanation:**
An idempotent matrix is one that satisfies the condition \( A^2 = A \). The goal is to demonstrate that for such a matrix \( A \), it must either be the identity matrix \( I \) or a singular matrix.
**Definitions:**
- **Idempotent Matrix**: A matrix \( A \) such that \( A^2 = A \).
- **Identity Matrix (\( I \))**: A square matrix with ones on the diagonal and zeros elsewhere, satisfying \( AI = IA = A \) for any matrix \( A \) of the same size.
- **Singular Matrix**: A matrix that does not have an inverse.
**Approach:**
To prove the statement, consider the eigenvalues of the idempotent matrix. An eigenvalue \( \lambda \) of an idempotent matrix will satisfy the equation:
\[
\lambda^2 = \lambda
\]
This simplifies to:
\[
\lambda(\lambda - 1) = 0
\]
Thus, \( \lambda \) can either be 0 or 1. This implies:
- If all eigenvalues are 1, \( A \) acts as the identity matrix.
- If any eigenvalue is 0, \( A \) is singular (since a singular matrix is one with determinant 0, which occurs if there’s an eigenvalue of 0).
This establishes that for idempotent \( A \), \( A = I \) or \( A \) is singular.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F410816ed-4be2-4291-b0fb-4c8f4b2206ab%2Fdc215cf3-110d-4054-a8cf-2b86824bd1ff%2Faum931_processed.png&w=3840&q=75)
Transcribed Image Text:**Proof of Idempotent Matrix Properties**
**Statement:**
Prove that if \( A \) is idempotent, i.e., that \( A^2 = A \), then either \( A = I \) or \( A \) is singular.
**Explanation:**
An idempotent matrix is one that satisfies the condition \( A^2 = A \). The goal is to demonstrate that for such a matrix \( A \), it must either be the identity matrix \( I \) or a singular matrix.
**Definitions:**
- **Idempotent Matrix**: A matrix \( A \) such that \( A^2 = A \).
- **Identity Matrix (\( I \))**: A square matrix with ones on the diagonal and zeros elsewhere, satisfying \( AI = IA = A \) for any matrix \( A \) of the same size.
- **Singular Matrix**: A matrix that does not have an inverse.
**Approach:**
To prove the statement, consider the eigenvalues of the idempotent matrix. An eigenvalue \( \lambda \) of an idempotent matrix will satisfy the equation:
\[
\lambda^2 = \lambda
\]
This simplifies to:
\[
\lambda(\lambda - 1) = 0
\]
Thus, \( \lambda \) can either be 0 or 1. This implies:
- If all eigenvalues are 1, \( A \) acts as the identity matrix.
- If any eigenvalue is 0, \( A \) is singular (since a singular matrix is one with determinant 0, which occurs if there’s an eigenvalue of 0).
This establishes that for idempotent \( A \), \( A = I \) or \( A \) is singular.
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