Advanced Linear Algebra - Lecture 16

Key Topics

• Square root of a linear transformation
• Transpose of a matrix
• Derivative map and non-integer powers

Diagonalization and Matrix Powers

• Diagonalization:
  • A matrix can be diagonalized if it can be written as (PDP^{-1}), where:
    • (D) is diagonal.
    • (P) is invertible, with columns as eigenvectors of the matrix.
    • Eigenvalues of the matrix are the entries of (D).
• Matrix Powers:
  • Integer powers (A^k = A \times A \times ... \times A (k \text{ times})).
  • Non-integer powers are defined using diagonalization: (A^r = PD^rP^{-1}).
  • Powers of diagonal matrices involve powering the diagonal entries.

Square Root of the Transpose Map

• Objective: Find a linear transformation (S) such that (S^2 = T), the transpose map.
• Method:
  • Diagonalize the transpose map using a known basis.
  • Take the square root of each diagonal entry.
  • Calculate (S), ensuring it represents (T^{1/2}).
• Result: A matrix representation of (S) is derived, allowing us to determine its action on 2x2 matrices.

Square Root of the Derivative Map

• Objective: Apply the concept of fractional calculus - find half a derivative of sine and cosine functions.
• Steps:
  • Convert differentiation into a matrix form using the basis ({\sin, \cos}).
  • Diagonalize this matrix; eigenvalues and vectors help achieve this.
  • Calculate the square root by diagonalizing and using powers on the diagonal matrix.
• Result: The resulting matrix gives a real output despite complex intermediate steps.

Generalizing to Non-integer Derivatives

• Rotation Matrix Concept:
  • The differentiation matrix ([0, -1, 1, 0]) is a rotation matrix (counterclockwise by 90 degrees).
  • Non-integer powers imply rotations by smaller angles (e.g., (\pi r/2)).
• Geometric Interpretation:
  • Understanding the matrix as a rotation simplifies finding any real number power.
  • For sin and cos, each derivative represents a phase shift by (\pi r/2).

Conclusion

• These methods highlight the power of diagonalization and linear algebra techniques in solving complex problems that seem counterintuitive.
• Applications span from linear transformations in matrix theory to fractional calculus in calculus.

This lecture demonstrates how foundational linear algebra concepts can be extended to solve intricate problems, showcasing their wide applicability and potential for innovative solutions.