6.1 Power Method

What is the power method

• building off 6.2 Diagonalization

  • it uses the same formula $P^{-1}AP=D$ to take advantage that we can use it too take powers of matrices with out too many calculations

• Definition

  • $A^k=P{D}^k{P}^{-1}$

the dominant eigenvalue

• what is the dominant eigen value

  • it’s the eigen value that has the greatest absolute value from the other eigen values of the vector

    • this also reaches to eigenspace (dominant eigenspace)

• dominant eigen space

  • how to find

    • let $A$ be the real dominant eigenvalue ${\lambda }_1$. and the initial vector be ${\vec{x}}_0$ calculate the power sequence
      • $A{\vec{x}}_0,A^2{\vec{x}}_0,A^3{\vec{x}}_0,\dots$
        • it turns out doing this converges to the dominant eigenspace

    • to describe this convergence we can use

      • ${\vec{x}}_1=\frac{A{\vec{x}}_0}{\mid \mid A{\vec{x}}_0\mid \mid },{\vec{x}}_2=\frac{A{\vec{x}}_1}{\mid \mid A{\vec{x}}_1\mid \mid },\dots$
        • this sequence will almost always converge to a dominant eigenvector $\vec{q}$ and $A\vec{q}={\lambda }_1\vec{q}$