Diagonalize the following matrix. – 13 - 42 5 16 %3D D P-1 =
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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![**Diagonalizing a Matrix**
In this example, we are tasked with diagonalizing the given matrix:
\[
\begin{bmatrix}
-13 & -42 \\
5 & 16
\end{bmatrix}
\]
To diagonalize the matrix, we need to find matrices \( P \) and \( D \) such that:
\[
A = PDP^{-1}
\]
**Matrices:**
1. **Matrix \( P \):**
- This is the matrix of eigenvectors of the original matrix.
- It is a 2x2 matrix shown here with placeholders for its components.
\[
P =
\begin{bmatrix}
\text{[ ]} & \text{[ ]} \\
\text{[ ]} & \text{[ ]}
\end{bmatrix}
\]
2. **Matrix \( D \):**
- This is the diagonal matrix containing the eigenvalues of the original matrix.
- It is also a 2x2 matrix with placeholders for the eigenvalues.
\[
D =
\begin{bmatrix}
\text{[ ]} & 0 \\
0 & \text{[ ]}
\end{bmatrix}
\]
3. **Matrix \( P^{-1} \):**
- This is the inverse of the matrix \( P \).
- It is represented with placeholders for its components.
\[
P^{-1} =
\begin{bmatrix}
\text{[ ]} & \text{[ ]} \\
\text{[ ]} & \text{[ ]}
\end{bmatrix}
\]
**Explanation:**
To diagonalize the matrix:
- Calculate the eigenvalues and eigenvectors of the original matrix.
- Construct the matrices \( P \) and \( D \) from these values.
- Verify the diagonalization by checking if \( PDP^{-1} \) reconstructs the original matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F200b2eb6-b630-49f4-95ef-5591909fdfea%2F6a5bb7e2-92fb-4050-a7b7-bc7a261086bc%2F6giosx_processed.png&w=3840&q=75)
Transcribed Image Text:**Diagonalizing a Matrix**
In this example, we are tasked with diagonalizing the given matrix:
\[
\begin{bmatrix}
-13 & -42 \\
5 & 16
\end{bmatrix}
\]
To diagonalize the matrix, we need to find matrices \( P \) and \( D \) such that:
\[
A = PDP^{-1}
\]
**Matrices:**
1. **Matrix \( P \):**
- This is the matrix of eigenvectors of the original matrix.
- It is a 2x2 matrix shown here with placeholders for its components.
\[
P =
\begin{bmatrix}
\text{[ ]} & \text{[ ]} \\
\text{[ ]} & \text{[ ]}
\end{bmatrix}
\]
2. **Matrix \( D \):**
- This is the diagonal matrix containing the eigenvalues of the original matrix.
- It is also a 2x2 matrix with placeholders for the eigenvalues.
\[
D =
\begin{bmatrix}
\text{[ ]} & 0 \\
0 & \text{[ ]}
\end{bmatrix}
\]
3. **Matrix \( P^{-1} \):**
- This is the inverse of the matrix \( P \).
- It is represented with placeholders for its components.
\[
P^{-1} =
\begin{bmatrix}
\text{[ ]} & \text{[ ]} \\
\text{[ ]} & \text{[ ]}
\end{bmatrix}
\]
**Explanation:**
To diagonalize the matrix:
- Calculate the eigenvalues and eigenvectors of the original matrix.
- Construct the matrices \( P \) and \( D \) from these values.
- Verify the diagonalization by checking if \( PDP^{-1} \) reconstructs the original matrix.
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