Invertible Matrix

If a square matrix A is invertible, then an Inverse Matrix of A exists (usually written A^{-1}) exists.

The multiplication of a matrix and its inverse results in the Identity Matrix:

A A^{-1} = A^{-1} A = I

Tutorials about invertible matrices (usually) only discuss scenarios where A is a Square Matrix.

We might say we are only interested in stories where both the vectors and one forms have n dimensions and the matrices are n x n, the rank three tensors are n x n x n, and so on…

Appendix A

We sometimes write subscripts for the numbers of rows and columns in a matrix. When we do this, the first number is the number of rows and the second number is the number of columns.

Suppose we have A_{np} then it is required that we have A^{-1}_{pn}. This is necessary to to get the row to column matching that we need to make matrix multiplication possible.

The above discussion only considers requirements for matrix multiplication. What else might there be, ergo, what could go wrong?

We found this note in a pirate’s handwriting…

A matrix represents a map and calling a matrix invertible means that the map is invertible and this is only possible if that map is an Isomorphism between Linear Spaces of the same dimension.

If our spaces are the Vector Spaces V and W and they are of the same dimension then we can let n be that value.

  • The mapping from V to W is an nxn matrix.
  • The mapping from W to V is an nxn matrix.