How Does Matrix Transposition Affect the Product A^T A?

[jtruth914]

Let $B=A^{T}A$. Show that $b_{ij}$=$a^{T}_{i}$$a_{j}$.

I have no idea how to approach this problem.

 

[Mandelbroth]

Let $B=A^{T}A$. Show that $b_{ij}$=$a^{T}_{i}$$a_{j}$.

I have no idea how to approach this problem.

I don't understand your notation. Could you please clarify what you've written?

 

[.Scott]

Let $B=A^{T}A$. Show that $b_{ij}$=$a^{T}_{i}$$a_{j}$.

When you transpose $A$You are flipping the rows and columns.
When you multiply $A^{T}A$ you would generate each element $b_{ij}$ will be the dot product of row i of the first matrix and column j of the second matrix. Since the first matrix is the transposition of the second, row i of that transposition will be column i of the original. So each element will be the dot product of two column vectors:

$b_{ij}$=$a_{i}$·$a_{j}$

Using matrix notation, this is:

$b_{ij}$=$a^{T}_{i}$$a_{j}$