Math Problem Statement
B = 1 0 0 0 1 0 0 0 −11 Find P such that P−1 c) · A · P = B.
Solution
To find a matrix such that , we need to diagonalize with eigenvalues forming the diagonal of . Here's the process:
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Matrix :
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -11 \end{pmatrix}$$ The eigenvalues of $$A$$ must match $$\lambda_1 = 1, \lambda_2 = 1, \lambda_3 = -11$$, since $$B$$ is the diagonalized form of $$A$$. -
Eigenvalues and Eigenvectors of :
- Compute the eigenvalues of .
- For each eigenvalue , compute the eigenvectors by solving .
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Constructing :
- The matrix is formed by arranging the eigenvectors of corresponding to as columns.
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Verification:
- Verify .
If you'd like, I can compute this explicitly for a given . Could you provide , or should I proceed with a symbolic approach?
Do you want me to solve this explicitly with a sample , or would you like the symbolic process laid out in detail?
Related Questions
- How do eigenvalues relate to diagonalization?
- What happens if is not diagonalizable?
- How is computed from ?
- What is the role of similarity transformations in linear algebra?
- Can a non-square matrix be diagonalized?
Tip
When diagonalizing, ensure that the eigenvectors of are linearly independent; otherwise, cannot be diagonalized.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Diagonalization
Eigenvalues and Eigenvectors
Similarity Transformations
Formulas
P^{-1} · A · P = B
(A - λI)·v = 0
Matrix inversion and computation of eigenvectors
Theorems
Eigenvalue Decomposition Theorem
Diagonalization Theorem
Similarity Transformation
Suitable Grade Level
Undergraduate Level (Linear Algebra Course)