Question 1: $A^*A=\sum_iA_i\otimes A_i$, why? I don't understand this tensor product notation.
Question 2: $A^*A=(\langle A_j,A_k\rangle)_{jk}$ if $A$ has independent columns, why? What does the right-hand-side notation mean? Why independent columns matters?
This comes from the book probability in high dimensions.
The first equality: Left hand side $\overline{\pmatrix{a&c\cr b&d}}\pmatrix{a&b\cr c&d}=\pmatrix{\overline{a}a+\overline{c}c&\overline{a}b+\overline{c}d\cr\overline{b}a+\overline{d}c&\overline{b}b+\overline{d}d}$. Right hand side $\pmatrix{a&b}\otimes\pmatrix{a&b}+\pmatrix{c&d}\otimes\pmatrix{c&d}=\pmatrix{a&b}^*\cdot\pmatrix{a&b}+\pmatrix{c&d}^*\cdot\pmatrix{c&d}=\overline{\pmatrix{a\cr b}}\cdot\pmatrix{a& b}+\overline{\pmatrix{c\cr d}}\pmatrix{c&d}=\pmatrix{a\overline{a}&\overline{a}b\cr \overline{b}a&b\overline{b}}+\pmatrix{c\overline{c}&\overline{c}d\cr \overline{d}c&d\overline{d}}=\pmatrix{\overline{a}a+\overline{c}c&\overline{a}b+\overline{c}d\cr\overline{b}a+\overline{d}c&\overline{b}b+\overline{d}d}$.
This is based on the definition of outer product of row vectors $v\otimes w=v^*\cdot w$
It looks to me more appropriate to say $(A^*A)_{jk}=\langle A_j,A_k\rangle$ and no need for independence.