Math Problem Statement

For any 3 by 3 matrix A, A and A(transpose) have the same set of eigenspaces

Solution

To determine whether AA and ATA^T (the transpose of AA) share the same eigenspaces, we need to explore the fundamental properties of eigenvalues and eigenspaces of square matrices.

Analysis:

  1. Eigenvalues and Eigenvectors:

    • For a matrix AA, if v\mathbf{v} is an eigenvector associated with the eigenvalue λ\lambda, then: Av=λv.A \mathbf{v} = \lambda \mathbf{v}.
  2. Transpose and Symmetry:

    • For the transpose matrix ATA^T, the eigenvalues are the same as those of AA. This can be shown by noting that det(AλI)=det(ATλI)\det(A - \lambda I) = \det(A^T - \lambda I), since the determinant of a matrix and its transpose are equal.
  3. Eigenspaces:

    • While AA and ATA^T have the same eigenvalues, the eigenspaces corresponding to a given eigenvalue may differ. This is because AA and ATA^T are not guaranteed to have the same eigenvectors unless AA is symmetric. If AA is symmetric, then A=ATA = A^T, and the eigenspaces will indeed be identical.
  4. General 3x3 Matrix:

    • For a general 3×33 \times 3 matrix AA, AA and ATA^T may have the same eigenvalues but may not have identical eigenspaces unless AA is symmetric.

Conclusion:

The statement "For any 3×33 \times 3 matrix AA, AA and ATA^T have the same set of eigenspaces" is false in general. They will only have the same eigenspaces if AA is symmetric.


Would you like more details or examples?
Here are 5 related questions to expand your understanding:

  1. What are the conditions under which a matrix is symmetric, and how does that impact eigenspaces?
  2. How can we verify whether AA and ATA^T share eigenvectors in practice?
  3. Can AA and ATA^T have different eigenvalues in any scenario? Why or why not?
  4. What is the relationship between the singular value decomposition (SVD) and eigenspaces?
  5. How do the eigenspaces of symmetric matrices differ from those of general matrices?

Tip: When studying eigenspaces, always consider the geometric multiplicity (dimension of eigenspace) and how it relates to the algebraic

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenspaces
Matrix Transpose
Symmetric Matrices

Formulas

A * v = λ * v (eigenvalue equation)
det(A - λI) = det(A^T - λI) (property of determinants)

Theorems

Symmetric Matrix Theorem (symmetric matrices have identical eigenspaces for A and A^T)
Determinant Equality (det(A) = det(A^T))

Suitable Grade Level

Undergraduate (Linear Algebra)