the matrix A = -5 1 3 -3 Find an invertible matrix S and a diagonal matrix D such that S-¹ AS = D. has eigenvalues A₁ = -6 and A2 = -2. The bases of the eigenspaces are v₁ = - [¹₁] and and V₂ = }] respectively. S= --188 D=
the matrix A = -5 1 3 -3 Find an invertible matrix S and a diagonal matrix D such that S-¹ AS = D. has eigenvalues A₁ = -6 and A2 = -2. The bases of the eigenspaces are v₁ = - [¹₁] and and V₂ = }] respectively. S= --188 D=
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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![### Matrix Diagonalization Problem
The matrix \( A = \begin{bmatrix} -5 & 1 \\ 3 & -3 \end{bmatrix} \) has eigenvalues \( \lambda_1 = -6 \) and \( \lambda_2 = -2 \).
The bases of the eigenspaces are:
\[ \mathbf{v_1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]
and
\[ \mathbf{v_2} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \]
respectively.
**Problem Statement**:
Find an invertible matrix \( S \) and a diagonal matrix \( D \) such that \( S^{-1}AS = D \).
**Solution Frame**:
To solve the problem, fill in the matrices \( S \) and \( D \) as shown below.
\[ S = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix} \]
\[ D = \begin{bmatrix} \boxed{} & 0 \\ 0 & \boxed{} \end{bmatrix} \]
Students are expected to input the correct eigenvectors into matrix \( S \) and the eigenvalues into matrix \( D \), then perform matrix multiplication to verify the condition \( S^{-1}AS = D \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2Fb3f201ca-33ab-4334-9e01-893b384935c8%2Fezvbhkl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Diagonalization Problem
The matrix \( A = \begin{bmatrix} -5 & 1 \\ 3 & -3 \end{bmatrix} \) has eigenvalues \( \lambda_1 = -6 \) and \( \lambda_2 = -2 \).
The bases of the eigenspaces are:
\[ \mathbf{v_1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]
and
\[ \mathbf{v_2} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \]
respectively.
**Problem Statement**:
Find an invertible matrix \( S \) and a diagonal matrix \( D \) such that \( S^{-1}AS = D \).
**Solution Frame**:
To solve the problem, fill in the matrices \( S \) and \( D \) as shown below.
\[ S = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix} \]
\[ D = \begin{bmatrix} \boxed{} & 0 \\ 0 & \boxed{} \end{bmatrix} \]
Students are expected to input the correct eigenvectors into matrix \( S \) and the eigenvalues into matrix \( D \), then perform matrix multiplication to verify the condition \( S^{-1}AS = D \).
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