Knowlpedia

Matrix representation

A matrix encoding a linear map relative to chosen bases.

The matrix representation of a T:V→WT: V \to W with respect to bases B={v1,…,vn}\mathcal{B} = \{v_1, \ldots, v_n\} of VV and C={w1,…,wm}\mathcal{C} = \{w_1, \ldots, w_m\} of WW is the m×nm \times n [T]BC[T]_{\mathcal{B}}^{\mathcal{C}} whose columns are the coordinate vectors of T(vj)T(v_j) in the basis C\mathcal{C}.

That is, if T(vj)=∑i=1maijwiT(v_j) = \sum_{i=1}^m a_{ij} w_i, then the (i,j)(i,j) entry of the matrix is aija_{ij}.

Writing FF for the field of scalars, the matrix is defined so that the square below commutes, where the vertical maps take coordinate vectors in the bases B\mathcal B and C\mathcal C:

Commutative diagramV \arrow[r, "T"] \arrow[d, "{[\,\cdot\,]_{\mathcal B}}"', "\cong"] & W \arrow[d, "{[\,\cdot\,]_{\mathcal C}}", "\cong"'] \\ F^n \arrow[r, "{[T]_{\mathcal B}^{\mathcal C}}"'] & F^m
Properties
  • Composition: [ST]BD=[S]CD[T]BC[ST]_{\mathcal{B}}^{\mathcal{D}} = [S]_{\mathcal{C}}^{\mathcal{D}} [T]_{\mathcal{B}}^{\mathcal{C}}.
  • Change of basis: If PP and QQ satisfy [v]B=P[v]B′[v]_{\mathcal B}=P[v]_{\mathcal B'} and [w]C=Q[w]C′[w]_{\mathcal C}=Q[w]_{\mathcal C'}, then [T]B′C′=Q−1[T]BCP[T]_{\mathcal{B}'}^{\mathcal{C}'} = Q^{-1} [T]_{\mathcal{B}}^{\mathcal{C}} P.
  • Similarity: Two matrices represent the same operator (different bases) iff they are similar.
Standard representation

For T:Rn→RmT: \mathbb{R}^n \to \mathbb{R}^m, the standard matrix is [T(e1)∣⋯∣T(en)][T(e_1) \mid \cdots \mid T(e_n)].

Invariants

The is independent of both basis choices. For an endomorphism T:V→VT:V\to V, the and are also independent of the basis used for both domain and codomain.