Linear Algebra for Machine Learning (4 of 4) – (The Flow of Tensors)

In this final and 4th Part of our brief look into Linear Algebra, we’ll talk about the Transpose and Inverse of matrices.

Matrix Transpose

In other words, $$ A_{ij} = (A^T)_{ji} $$

Think of it like rotating the matrix 90° in clockwise direction and then flipping it.

For example, $$ A = \begin{bmatrix} 2&3&2&5 \\ 1&3&4&5 \end{bmatrix} $$ $$ A^T = \begin{bmatrix} 2&1 \\ 3&3 \\ 2&4 \\ 5&5 \end{bmatrix} $$

Matrix Inverse

A square (matrix A) is invertible if there exist a (matrix B) such that (AB = I) and (BA = I)

The inverse of a matrix is denoted by (A^{-1})

For example, given (A = \begin{bmatrix} 2&7 \\ 1&4 \end{bmatrix} ) and (B = \begin{bmatrix} 4&-7 \\ -1&2 \end{bmatrix} )

We have (A*B) = ( \begin{bmatrix} 2&7 \\ 1&4 \end{bmatrix} \begin{bmatrix} 4&-7 \\ -1&2 \end{bmatrix}
= \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} = I )

The existence of (B) makes (A) invertible. Same can also be said vice versa.

Note, however, that a non-square matrix does not have an inverse matrix.

With these, we’ve learnt the basics of the arithmetic of tensors.

We’ll dive straight into TensorFlow in the next tutorial.