Question

Show A^T * A = A*A^T.

Answer 1

\[a^t*a=a*a^t\]Some type of property referring to the order of multiplications terms?

Answer 2

I think that T is the transpose of a matrix.

Answer 3

I don't understand. I'm supposed to show that it equals I2, the identity matrix for a 2 x 2 matrix.

Answer 4

I know the identity matrix for a 2 x 2 matrix is (10...01)

Answer 5

Matrix multiplication is not commutative. @Directrix

Answer 6

@amaes Yes, I agree. There are no instructions with the problem. Hence, we took the symbols to be garden variety algebraic variables.

Answer 7

Okay.

Answer 8

\[\text{Let }A=\left[\begin{matrix}a_{1,1}&a_{1,2}\\
a_{2,1}&a_{2,2}\end{matrix}\right]\text{ and }A^T=\left[\begin{matrix}a_{2,2}&a_{2,1}\\
a_{1,2}&a_{1,1}\end{matrix}\right]\]
Now,
\[A\cdot A^T=\left[\begin{matrix}a_{1,1}\;a_{2,2}+(a_{1,2})^2&a_{1,1}\;a_{2,1}+a_{1,2}\;a_{1,1}\\
a_{2,1}\;a_{2,2}+a_{2,2}\;a_{1,2}&(a_{2,1})^2+a_{2,2}\;a_{1,1}\end{matrix}\right]\]
Taking advantage of commutativity over multiplication, you can easily decompose this matrix to get A^T∙A.