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 AYou 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}