Let omega4 epi2 i Show that omega4 4 1 and 1 omegak 4 om

Let omega_4 = e^-pi/2 i. Show that omega^4 _4 = 1 and 1 + omega^k _4 + (omega^k _4)^2 + (omega^k _4)^3 = 0 for any k. Show that the columns of A are orthogonal to each other A = (1 1 1 1 1 omega_4 omega^2 _4 omega^3 _4 1 omega^2 _4 omega^4 _4 omega^6 _4 1 omega^3 _4 omega^6 _4 omega^9 _4).

Solution

As we know that a matrix is orthogonal if

A^-1=A^T

i.e., AA^T = I

we can clearly see that the transpose of A is equal to A

i.e., A.A^T=A²

we have to show that A²=I

and then we can show that the column of A are orthogonal to each other.

a square real matrix with orthonormal columns is called orthogonal

if A is orthogonal, then

A is invertible, with inverse A^T : A^TA = I

A is square = AA^T = I

• A^T is also an orthogonal matrix

• columns of A are orthonormal