Question

Linear algebra fun

Answer 1

What are all the matrices A with real entries that satisfy this equation:

\[A^TA=0\]

Answer 2

\((A^TA)_{11} = \sum\limits_{k}A^T_{1k}A_{k1} = \sum\limits_{k}A_{k1}^2\)

that equals \(0\) only when the entire column is 0
so we may conclude \(A = 0\)

Answer 3

The proof I came up with is as follows:

Transform an arbitrary vector by A,

\[u=Av\]

What is the length of u?

\[u^\top u = v^\top A^\top A v = 0\]

So it must be that A is the zero matrix since it turns every arbitrary vector v into a vector of length 0!