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 PP such that P1AP=BP^{-1} \cdot A \cdot P = B, we need PP to diagonalize AA with eigenvalues forming the diagonal of BB. Here's the process:

  1. Matrix BB:

    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$$.
  2. Eigenvalues and Eigenvectors of AA:

    • Compute the eigenvalues of AA.
    • For each eigenvalue λi\lambda_i, compute the eigenvectors by solving (AλiI)v=0(A - \lambda_i I) \vec{v} = 0.
  3. Constructing PP:

    • The matrix PP is formed by arranging the eigenvectors of AA corresponding to λ1,λ2,λ3\lambda_1, \lambda_2, \lambda_3 as columns.
  4. Verification:

    • Verify P1AP=BP^{-1} A P = B.

If you'd like, I can compute this explicitly for a given AA. Could you provide AA, or should I proceed with a symbolic approach?


Do you want me to solve this explicitly with a sample AA, or would you like the symbolic process laid out in detail?

Related Questions

  1. How do eigenvalues relate to diagonalization?
  2. What happens if AA is not diagonalizable?
  3. How is P1P^{-1} computed from PP?
  4. What is the role of similarity transformations in linear algebra?
  5. Can a non-square matrix AA be diagonalized?

Tip

When diagonalizing, ensure that the eigenvectors of AA are linearly independent; otherwise, AA 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)