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[61] Diagonalize B = (by finding P and D) or explain why B isn't diagonalizable.

[61] Diagonalize B = (by finding P and D) or explain why B isn't diagonalizable.

Advanced Engineering Mathematics
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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**Problem: Diagonalize the Matrix B** Given the matrix \( B = \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} \), our objective is to either: 1. Find matrices \( P \) and \( D \) such that \( B = PDP^{-1} \), where \( D \) is a diagonal matrix, and \( P \) is an invertible matrix, or 2. Explain why \( B \) is not diagonalizable. To find \( P \) and \( D \): - **Step 1**: Calculate the eigenvalues of matrix \( B \). - **Step 2**: Determine the eigenvectors corresponding to each eigenvalue. - **Step 3**: Form matrix \( P \) using the eigenvectors as its columns. - **Step 4**: Construct the diagonal matrix \( D \) with the eigenvalues on its main diagonal. If there aren't enough linearly independent eigenvectors to form \( P \), then matrix \( B \) is not diagonalizable.
Transcribed Image Text:**Problem: Diagonalize the Matrix B** Given the matrix \( B = \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} \), our objective is to either: 1. Find matrices \( P \) and \( D \) such that \( B = PDP^{-1} \), where \( D \) is a diagonal matrix, and \( P \) is an invertible matrix, or 2. Explain why \( B \) is not diagonalizable. To find \( P \) and \( D \): - **Step 1**: Calculate the eigenvalues of matrix \( B \). - **Step 2**: Determine the eigenvectors corresponding to each eigenvalue. - **Step 3**: Form matrix \( P \) using the eigenvectors as its columns. - **Step 4**: Construct the diagonal matrix \( D \) with the eigenvalues on its main diagonal. If there aren't enough linearly independent eigenvectors to form \( P \), then matrix \( B \) is not diagonalizable.
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