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Diagonalization

Similar matrices

2 square matrices and of the same order, are similar iff there exists an invertible matrix such that:

Similarity of 2 matrices is commutative.

Similar matrices have the set of eigenvalues.

Definition

A matrix is diagonalizable if it is similar to a diagonal matrix.

Here:

  • is a diagonal matrix
  • is an invertible matrix

Steps

  • Find eigenvalues of :
  • Find corresponding eigenvectors:
  • Construct by joining the eigenvectors as columns

Uses

Finding integer powers

Suppose is diagonalizable, and .