Math Problem Statement

How do you diagonalize the given matrix A using orthogonal transformations?

Solution

The image appears to show a handwritten note on diagonalizing a matrix using orthogonal transformations. Here's a breakdown of what's shown:

  1. Introduction to Orthogonal Transformations:

    • An orthogonal transformation transforms a matrix into a diagonal matrix using the relation NTANN^T A N, where NN is an orthogonal matrix.
    • A matrix AA is orthogonal if A1=ATA^{-1} = A^T.
    • Properties: AAT=ATA=IAA^T = A^T A = I, where II is the identity matrix.
  2. Matrix to Diagonalize:

    3 & 1 & 1 \\ 1 & 3 & -1 \\ 1 & -1 & 3 \end{pmatrix}$$ This matrix needs to be diagonalized.
  3. Eigenvalues λ\lambda:

    • λ1=1\lambda_1 = 1, λ2=4\lambda_2 = 4, λ3=4\lambda_3 = 4.
  4. Eigenvectors:

    • For λ1=1\lambda_1 = 1: x1=(110)x_1 = \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}
    • For λ2=4\lambda_2 = 4: x2=(111)x_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}
    • For λ3=4\lambda_3 = 4: x3=(112)x_3 = \begin{pmatrix} -1 \\ 1 \\ -2 \end{pmatrix}
  5. Diagonalized Matrix: After applying the orthogonal transformation, the resulting diagonal matrix will have the eigenvalues as its diagonal entries.

  6. Additional Matrices: There are additional matrices provided with different eigenvalues (like λ=0,1,3\lambda = 0,1,3 and λ=1,2,2\lambda = -1,2,2) that seem to be exercises or homework tasks for further diagonalization by orthogonal transformations.

Would you like a detailed step-by-step explanation of diagonalization, or help with a specific part of this image?

5 Related Questions:

  1. What is the process to find eigenvalues for a given matrix?
  2. How are eigenvectors derived for each eigenvalue?
  3. What properties does an orthogonal matrix have?
  4. How do you confirm that a matrix is diagonalizable?
  5. Can all square matrices be diagonalized using orthogonal transformations?

Tip:

When diagonalizing matrices, always check if the matrix is symmetric. Symmetric matrices are always diagonalizable using orthogonal matrices, simplifying the process!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Diagonalization
Eigenvalues and Eigenvectors
Orthogonal Transformations

Formulas

Eigenvalue Equation: det(A - λI) = 0
Orthogonality Condition: A^(-1) = A^T

Theorems

Spectral Theorem
Orthogonal Diagonalization Theorem

Suitable Grade Level

College Level