Change of Basis

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Square Matrices

Definition 1 An \(I\times I\) matrix is called a square matrix. We denote the collection of all such matrices by \(\Mat_I(A)\).

In particular, when \(I\) is finite and \(A\) is commutative, \(\Mat_{n}(A)\) has additional structure: it is not merely an \(A\)-module, but also carries a multiplication. That is, \(\Mat_{n}(A)\) is an \(A\)-algebra.

Proposition 2 In this situation, \(\Mat_n(A)\) is a unital associative algebra.

Proof

That \(\Mat_n(A)\) is an associative \(A\)-algebra follows immediately from §Matrices, §§Matrix Multiplication. The identity element for multiplication in \(\Mat_n(A)\) is the identity matrix

\[I_n=\begin{pmatrix}1&0&\cdots&0\\0&1&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\cdots&1\end{pmatrix}\]

as is readily verified.

\(M_n(A)\) has the canonical basis \((E_{ij})\), and the structure constants with respect to this basis are given by

\[E_{ij}E_{jk}=\delta_{jh}E_{ik}\]

Definition 3 We denote by \(\GL_n(A)\) the subset of \(\Mat_n(A)\) consisting of those elements that admit a multiplicative inverse.

Fix a basis \(\mathcal{B}=(e_i)_{i\in I}\) of a free \(A\)-module \(M\), and let \(\lvert I\rvert=n\). Then for any \(u\in \End_{\lMod{A}}(M)\), we have \([u]_{\mathcal{B}}^\mathcal{B}\in\Mat_n(A)\); moreover, if \(u\) is an isomorphism, then by §Matrices and Linear Maps, ⁋Corollary 4 we have \([u]_{\mathcal{B}}^\mathcal{B}\in\GL_n(A)\). In this case, §Dual Spaces, ⁋Proposition 5 and §Matrices and Linear Maps, ⁋Proposition 5 yield the identity

\[\bigl([u^{-1}]_{\mathcal{B}}^\mathcal{B}\bigr)^t=\bigl(\bigl[u^\ast\bigr]_{\mathcal{B}^\ast}^{\mathcal{B}^\ast}\bigr)^{-1}\]

Change of Basis

Proposition 4 Let \(M\) be an arbitrary \(A\)-module and let \(\mathcal{B}=(e_i)_{i\in I}\) be a finite basis of \(M\). Then the formula

\[e_i'=\sum_{j=1}^n a_{ji}e_i,\qquad 1\leq i\leq n\]

defines a basis of \(M\) if and only if the square matrix \((a_{ji})\) is invertible.

Proof

The given matrix \((a_{ji})\) is precisely the matrix representation \([u]_{\mathcal{B}}^\mathcal{B}\in\Mat_n(A)\) with respect to \(\mathcal{B}\) of the linear map \(u\in\End_{\lMod{A}}(M)\) defined by

\[u:e_i\mapsto e_i'=\sum_{j=1}^n a_{ji}e_i\]

This matrix is invertible if and only if \(u\) is an isomorphism, which in turn holds if and only if \((u(e_i))_{i\in I}\) is a basis of \(M\).

Conversely, we may also regard the matrix \((a_{ji})\) as the matrix representation of the identity map \(\id_M:M \rightarrow M\) with respect to different bases. Write the basis \((e_i')\) as \(\mathcal{B}'\). Then

\[\id_M(e_i')=\sum_{j=1}^n a_{ji}e_i\]

and thus

\[([\id_M]^{\mathcal{B}'}_\mathcal{B})=(\langle \id_M(e_i'), e_j^\ast\rangle)_{(j,i)\in J\times I}=(a_{ji})_{(j,i)\in J\times I}\]

From this perspective, this matrix is also called the change-of-basis matrix from \(\mathcal{B}'\) to \(\mathcal{B}\).

More generally, we have the following.

Proposition 5 Let \(M\) and \(N\) be \(A\)-modules with finite bases \(\mathcal{B}=(e_i)_{i\in I}\) and \(\mathcal{C}=(f_j)_{j\in J}\), respectively. For other bases \(\mathcal{B}'=(e_i')_{i\in I}\) of \(M\) and \(\mathcal{C}'=(f_j')_{j\in J}\) of \(N\), the formula

\[[u]_{\mathcal{C}'}^{\mathcal{B}'}=[\id_N]^\mathcal{C}_{\mathcal{C}'}[u]^\mathcal{B}_\mathcal{C}[\id_M]^{\mathcal{B}'}_{\mathcal{B}}\]

holds.

Similar Matrices

Definition 6 Two \(m\times n\) matrices \(X, X'\) are called equivalent if there exist square matrices \(P\in\GL_m(A)\) and \(Q\in\GL_n(A)\) such that \(X'=PXQ\).

In the same context as the discussion preceding §Change of Basis, ⁋Definition 6, it is preferable to consider the following finer equivalence relation rather than mere equivalence of matrices.

Definition 7 Two \(n\times n\) matrices \(X, X'\) are called similar if there exists a square matrix \(P\in\GL_n(A)\) such that \(X'=PXP^{-1}\).

Then, setting \(M=N\), \(\mathcal{B}=\mathcal{C}\), and \(\mathcal{B}'=\mathcal{C}'\) in Proposition 5 above, we see that the matrix representations of an element \(u\in\End_\rMod{A}(M)\) with respect to different bases are similar to one another.