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Is there only one identity matrix $\begin{pmatrix} 1&0&...&...&0\\0&1&0&...&0\\...&0&1&...&0\\...&...&0&1&0\\...&...&...&0&1\end{pmatrix}$ etc.. Or are there different identity matrices for other bases?

A textbook example asks if $[T]_{\beta} = I$ (the $n\times n$ identity matrix) for some basis $\beta$, is $T$ the identity operator?

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    Imray, that's a *procedure*; I asked whether you know a *formula*. That is, if you have an operator $T$, and a basis $\alpha$, and another basis $\beta$, do you know a formula relating $[T]_{\alpha}$ and $[T]_{\beta}$?2012-10-12

2 Answers 2

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Suppose we want $ \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ to be true regardless of which matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is, so that $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is an identity matrix. Since it's true regardless of which matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is, it must be true in particular if $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, so we have $ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} =\begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. $ This last equality clearly implies that $\begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Conclusion: if $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is an identity matrix, then $\begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Therefore there is only one $2\times2$ identity matrix. And the same argument works for bigger matrices.

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By definition, $[Tv]_\beta = [T]_\beta [v]_\beta = [v]_\beta$ for all $v$. Therefore $Tv=v$ for all $v$, i.e. $T$ is the identity on $V$. But to address your real question, here are some more questions for you:

  • What happens if I try to perform a "change of basis" on the identity matrix?
  • What happens if I try to represent the identity operator (on some $V$) in some basis? What does the identity operator do to the basis vectors?
  • What is the definition of the identity matrix, and what should this have to do with bases for vector spaces?
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    Yes,$I$mean that if$I$have a $2 \times 2$ identity matrix, which means it was derived by multiplying 1 to the first and fourth component of the standard ordered $2 \times 2$ basis, and I change the matrices from standard basis to something funnier, the matrix identity changes.2012-10-12