Change Of Basis

Definition

similarity transformation relates two matrices $A$ and $B$ that represent the same linear operator under different bases. $B$ is similar to $A$ if there exists an invertible matrix $M$ such that: $B = M^{-1}AM$ where $M$ is the change-of-basis matrix.

• How to read: “The matrix B equals M inverse times A times M.”
• Meaning: $A$ and $B$ describe the same linear transformation in two different coordinate systems; $M$ converts between them.

Why It Matters

It allows us to view a problem from the most “transparent” mathematical perspective, simplifying complex transformations into their most intuitive forms.

Core Concepts

• Invariants: Similar matrices share the same eigenvalues, determinant, trace, and rank.
• Coordinate Vectors: If $[v]_V$ is the coordinate vector in basis $V$ and $[v]_W$ in basis $W$, then $[v]_W = M^{-1}[v]_V$.
  • How to read: “The coordinate vector v with respect to basis W equals M inverse times the coordinate vector v with respect to basis V.”
  • Meaning / when to use: To express the same geometric vector in a new basis, multiply by the change-of-basis matrix (or its inverse, depending on direction).
• Diagonalization: A matrix is diagonalizable if it is similar to a diagonal matrix $\Lambda$, i.e., $\Lambda = S^{-1}AS$, where $S$ is the matrix of eigenvectors.
  • How to read: “The matrix lambda equals S inverse times A times S.”
  • Meaning: In the eigenvector basis, the transformation acts by simple scaling along each axis—powers and exponentials of $A$ become trivial.
• Unitary Similarity: If $M$ is a unitary matrix $U$ (where $U^H U = I$), the transformation $U^{-1}AU = U^H AU$ preserves lengths and angles (e.g., Spectral Theorem for Hermitian matrices).
  • How to read: “The matrix U Hermitian times U equals the identity matrix I.”
  • Meaning: $U$ is an isometry (rotation/reflection in complex space); similarity by $U$ changes coordinates without distorting geometry.

Connected Concepts

• Linear Transformation
• Hermitian Matrices
• Jordan Form
• Fast Fourier Transform