If A is diagonalizable (i.e. C CAC = D), then calculate A*.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Diagonalization and Calculation of Powers of a Matrix**
If a matrix \( A \) is diagonalizable, meaning there exists a matrix \( C \) such that \( C^{-1}AC = D \), where \( D \) is a diagonal matrix, then calculate \( A^k \).
**Explanation:**
1. **Diagonalization**: A matrix \( A \) is said to be diagonalizable if it can be expressed in the form \( A = CDC^{-1} \), where \( C \) is an invertible matrix and \( D \) is a diagonal matrix.
2. **Calculating Powers of a Matrix**: Once \( A \) is expressed in its diagonal form:
\[
A^k = (CDC^{-1})^k = CD^kC^{-1}
\]
Here, \( D^k \) can be easily computed because \( D \) is a diagonal matrix. The entries of \( D^k \) are simply the entries of \( D \) raised to the power \( k \).
This method simplifies the computation of powers of matrices by utilizing the properties of diagonal matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9496fb66-15f0-4d2e-bc7e-ca3bb9b70577%2Fbbcaf31b-4340-4803-a917-8f6c2a49487e%2Fabs1wm_processed.png&w=3840&q=75)
Transcribed Image Text:**Diagonalization and Calculation of Powers of a Matrix**
If a matrix \( A \) is diagonalizable, meaning there exists a matrix \( C \) such that \( C^{-1}AC = D \), where \( D \) is a diagonal matrix, then calculate \( A^k \).
**Explanation:**
1. **Diagonalization**: A matrix \( A \) is said to be diagonalizable if it can be expressed in the form \( A = CDC^{-1} \), where \( C \) is an invertible matrix and \( D \) is a diagonal matrix.
2. **Calculating Powers of a Matrix**: Once \( A \) is expressed in its diagonal form:
\[
A^k = (CDC^{-1})^k = CD^kC^{-1}
\]
Here, \( D^k \) can be easily computed because \( D \) is a diagonal matrix. The entries of \( D^k \) are simply the entries of \( D \) raised to the power \( k \).
This method simplifies the computation of powers of matrices by utilizing the properties of diagonal matrices.
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