Let -8 0.5] -8 -3 Find an invertible matrix X and a diagonal matrix D such that X-¹AX = D. 1.88 88 A = X = D=
Let -8 0.5] -8 -3 Find an invertible matrix X and a diagonal matrix D such that X-¹AX = D. 1.88 88 A = X = 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 Exercise
Given matrix:
\[A = \begin{bmatrix} -8 & 0.5 \\ -8 & -3 \end{bmatrix}\]
Find an invertible matrix \(X\) and a diagonal matrix \(D\) such that:
\[X^{-1}AX = D\]
Fill in the appropriate values for matrices \(X\) and \(D\) below:
\[X = \begin{bmatrix}
\Box & \Box \\
\Box & \Box
\end{bmatrix}\]
\[D = \begin{bmatrix}
\Box & 0 \\
0 & \Box
\end{bmatrix}\]
In this exercise, we look to diagonalize the given matrix \(A\). This involves finding an invertible matrix \(X\) and a diagonal matrix \(D\) such that the similarity transformation \(X^{-1}AX\) yields the diagonal matrix \(D\). This is a common problem in linear algebra that helps in simplifying matrix computations, such as computing matrix powers.
Hints:
- To find \(X\) and \(D\), you may need to compute the eigenvalues and eigenvectors of the matrix \(A\).
- The diagonal elements of \(D\) are the eigenvalues of \(A\).
- The columns of \(X\) are the corresponding eigenvectors of \(A\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80624252-1099-44a7-a129-6e4dc1a5c385%2Fccceda89-98c9-40de-830b-f654b1fdf8c4%2Fwcd4rbr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Diagonalization Exercise
Given matrix:
\[A = \begin{bmatrix} -8 & 0.5 \\ -8 & -3 \end{bmatrix}\]
Find an invertible matrix \(X\) and a diagonal matrix \(D\) such that:
\[X^{-1}AX = D\]
Fill in the appropriate values for matrices \(X\) and \(D\) below:
\[X = \begin{bmatrix}
\Box & \Box \\
\Box & \Box
\end{bmatrix}\]
\[D = \begin{bmatrix}
\Box & 0 \\
0 & \Box
\end{bmatrix}\]
In this exercise, we look to diagonalize the given matrix \(A\). This involves finding an invertible matrix \(X\) and a diagonal matrix \(D\) such that the similarity transformation \(X^{-1}AX\) yields the diagonal matrix \(D\). This is a common problem in linear algebra that helps in simplifying matrix computations, such as computing matrix powers.
Hints:
- To find \(X\) and \(D\), you may need to compute the eigenvalues and eigenvectors of the matrix \(A\).
- The diagonal elements of \(D\) are the eigenvalues of \(A\).
- The columns of \(X\) are the corresponding eigenvectors of \(A\).
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