Two square matrices are said to be similar if they represent the same map under two possibly different bases.

Formally two square matrices and of size are called similar if there exists an invertable matrix such that

Properties

Consider the matrices , , and .

• Reflexivity: is similar to itself.
• Symmetry: If is similar to then is similar to .
• Transitivity: If is similar to and is similar to then is similar to .

Additionally if and are similar to each other they will have the same:

• Characteristic Polynomial
  • Eigenvalues
  • Determinant
  • Trace