What does the set of rows is orthonormal means?

Question

I have this problem:

What does the set of rows of the given matrix is orthonormal means? especially that our matrix is not a square matrix and so it is not an orthogonal matrix. Could anyone help me in this please?

Answer 1

Recall that $(A^TA)^2 = A^TAA^TA = A^T(AA^T)A$

Now, let us inspect $AA^T$ in more detail.

Recall that the definition of matrix multiplication is that the $i$'th row $j$'th column entry of a matrix $(AB)$ is $(AB)_{i,j}=\sum\limits_k (A)_{i,k}(B)_{k,j}$. That is, reworded, the $i$'th row $j$'th column entry of the product of two matrices is the dot product of the corresponding $i$'th row from the matrix on the left with the corresponding $j$'th column from the matrix on the right.

Now, recognize that we were told that the rows of $A$ are orthonormal and the columns of $A^T$ are just the rows of $A$ transposed.

As such, it follows that the $i$'th row $j$'th column entry of $(AA^T)$ will be the dot product of the $i$'th and $j$'th rows of $A$, which because the rows are orthonormal results in $1$ in the case that $i=j$ and results in $0$ otherwise by the definition of what it means to be orthonormal. That is to say, we will have $1$'s along the diagonal and $0$'s everywhere else. We finally recognize this result as simply being the identity matrix.

This gives us that $(A^TA)^2 = A^T(AA^T)A = A^TIA = A^TA$, proving the claim.