Identity matrix for non-square matrices

I am aware that this might be a trivial question, but is there an identity matrix I which has dimensions $$m \times m$$ for which IA=A for all matrices A with dimensions $$m \times n$$? I know that for square matrices, it is 1 along a diagonal and 0 for the other entries, but for non-square matrices this will be different. Thanks.

• If $m \neq n$ then you cannot even multiply two $m \times n$ matrices. Commented May 19 at 5:47
• but then is there a matrix I for which IA=A, where A is a m by n matrix Commented May 19 at 6:18
• The $m \times m$ identity matrix works here. Commented May 19 at 6:33

The identity matrix that is multiplication-compatible with the given matrix will work. The reason is simple, and I will illustrate this with an example. Say $$m =2$$ and $$n = 3$$. Consider the product $$AB$$ where $$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix}.$$ The product $$AB$$ can be viewed as $$A$$ acting on each column of $$B$$, as follows: $$AB = A \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} = \begin{bmatrix} A \begin{bmatrix} b_{11} \\ b_{21} \end{bmatrix} & A \begin{bmatrix} b_{12} \\ b_{22} \end{bmatrix} & A \begin{bmatrix} b_{13} \\ b_{23} \end{bmatrix} \end{bmatrix}.$$ Then, it is clear that the matrix $$A$$ that sends each vector to itself (and hence $$B$$ to itself), is the identity $$\mathbf{I}_{m}$$.
The key is to realize that the left-multiplication of $$A$$ with $$B$$ is the same as $$A$$ acting separately on the columns on $$B$$.