Lecture 11: Introductory Linear Algebra

Instructor

• Nathan Johnston

Topics Covered

1. Matrix Powers

   • Concept of matrix powers is analogous to powers of real numbers.
   • Denoted as (A^k), meaning matrix A multiplied by itself k times.
   • Example: (A^2 = A \times A), (A^3 = A \times A \times A).

2. Special Case: Power of Zero

   • (A^0) is defined as the identity matrix.
   • Analogous to (x^0 = 1) for real numbers.
   • Ensures desirable properties of matrix exponentiation.

3. Properties of Matrix Powers

   • For a square matrix A and non-negative integers k, r:
     • (A^k \times A^r = A^{k+r})
     • ((A^k)^r = A^{kr})
   • These properties mirror those of exponentiation for real numbers.

4. Limitations of Matrix Powers

   • Rules with different matrices in the base:
     • ((A \times B)^k \neq A^k \times B^k)
   • Non-commutativity of matrix multiplication prevents direct application of some real number exponentiation rules.
   • Matrix powers only make sense for square matrices.

5. Computing Matrix Powers

   • Example calculations with a 2x2 matrix:
     • Compute (A^2) using matrix multiplication.
     • Compute (A^3 = A \times A^2) or (A^3 = A^2 \times A).
   • Matrix multiplication properties allow for commutative operations when same matrix is involved.

6. Efficient Computation of Large Powers

   • Avoid sequential multiplication for large exponents.
   • Use Exponentiation by Squaring:
     • Example: (A^8 = (A^4)^2).
     • Reduces number of multiplications needed.

7. Future Topics

   • Generalizing matrix powers beyond non-negative integers:
     • Negative, fractional, and irrational exponents.
     • Concepts like (A^{-3}), (A^{\sqrt{2}}), (A^{\pi}), etc.

Upcoming

• Next lecture will cover block matrices and further matrix topics.

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Key Takeaways:

• Understanding of matrix powers analogous to powers of real numbers.
• Matrix exponentiation rules similar but not always identical to those for real numbers.
• Square matrices are necessary for matrix powers.
• Efficient computation techniques for large powers.
• Future lectures will expand on these concepts with more complex exponential definitions.