Change of Basis

Square Matrices

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

In particular, when \(I\) is a finite set and \(A\) is commutative, \(\Mat_{n}(A)\) has special properties: not only is this object an \(A\)-module, but it also has a multiplication defined on it. That is, \(\Mat_{n}(A)\) is an \(A\)-algebra.

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

Proof

That \(\Mat_n(A)\) is an associative \(A\)-algebra was already verified in §Matrices, §§Matrix Multiplication. One can verify that the identity element for multiplication on \(\Mat_n(A)\) is the following identity matrix:

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

\(\Mat_n(A)\) has a canonical basis \((E_{ij})\); considering their structure constants, we can write the following relation:

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

Definition 3 Among the elements of \(\Mat_n(A)\), the collection of those that have an inverse with respect to multiplication is denoted by \(\GL_n(A)\).

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

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

holds.

Change of Basis

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

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

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

Proof

The given matrix \((a_{ji})\) is, via the formula

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

defining the linear map \(u\in\End_{\lMod{A}}(M)\), precisely its matrix representation with respect to \(\mathcal{B}\), given by \([u]_{\mathcal{B}}^\mathcal{B}\in\Mat_n(A)\). Now, this matrix having an inverse is equivalent to \(u\) being an isomorphism, which in turn is equivalent to \((u(e_i))_{i\in I}\) being a basis of \(M\).

Conversely to the above proof, the matrix \((a_{ji})\) can also be thought of as the matrix representation of the identity map \(\id_M:M \rightarrow M\) with respect to two different bases. Let us write the basis \((e_i')\) as \(\mathcal{B}'\). Then, since

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

, we have

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

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

More generally, the following holds.

Proposition 5 Let two \(A\)-modules \(M,N\) and their respective finite bases \(\mathcal{B}=(e_i)_{i\in I}\) and \(\mathcal{C}=(f_j)_{j\in J}\) be given, and let \(u:M \rightarrow N\) be an \(A\)-linear map. For other bases of \(M\) and \(N\), denoted by \(\mathcal{B}'=(e_i')_{i\in I}\) and \(\mathcal{C}'=(f_j')_{j\in J}\) respectively, the equation

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

holds.

Similar Matrices

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

In the same vein as the discussion preceding Definition 6, it is better to consider the following equivalence relation, which is finer than that of equivalent matrices.

Definition 7 Two \(n\times n\) matrices \(X, X'\) are said to be 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 for an element of \(\End_{\lMod{A}}(M)\), \(u\), its matrix representations with respect to different bases are similar to each other.

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References

[Bou] Bourbaki, N. Algebra I. Elements of Mathematics. Springer. 1998.