For the orthogonal matrix
A = 1/sqrt(2) -1/sqrt(2)
-1/(sqrt(2)) -1/sqrt(2)
verify that (Ax,Ay)=(x,y) for any vectors x and y in R2.
Can someone please explain this
^ btw that matrix has 4 values in it.
To verify that (Ax, Ay) = (x,y) for any vectors x and y in R2, we need to calculate the inner product of the vectors Ax and Ay and compare it to the inner product of the vectors x and y.
Let's start by calculating Ax and Ay:
Ax = A * x = 1/sqrt(2) * x - 1/sqrt(2) * y
Ay = A * y = -1/sqrt(2) * x - 1/sqrt(2) * y
Next, let's calculate the inner product (dot product) of Ax and Ay:
(Ax, Ay) = (1/sqrt(2) * x - 1/sqrt(2) * y, -1/sqrt(2) * x - 1/sqrt(2) * y)
= (1/sqrt(2) * x, -1/sqrt(2) * x) + (-1/sqrt(2) * y, -1/sqrt(2) * y)
= (1/sqrt(2) * x, -1/sqrt(2) * x) + (-1/sqrt(2) * y, -1/sqrt(2) * y)
= (1/sqrt(2) * x, -1/sqrt(2) * x) + (-1/sqrt(2) * y, -1/sqrt(2) * y)
= [(1/sqrt(2))^2 * (x, x)] + [(1/sqrt(2))^2 * (y, y)]
= (1/2 * (x, x)) + (1/2 * (y, y))
= 1/2 * (x, x + y)
Now, let's calculate the inner product of x and y:
(x, y) = (x, y)
Since (Ax, Ay) = 1/2 * (x, x + y) and (x, y) = (x, y), for them to be equal, we have to show that (1/2) * (x, x + y) = (x, y).
To prove this, let's simplify:
(1/2) * (x, x + y) = (1/2) * (x, x) + (1/2) * (x, y)
= 1/2 * (x, x) + 1/2 * (y, x)
= 1/2 * [(x, x) + (y, x)]
= 1/2 * (x + y, x)
= (x + y, x) / 2
Finally, we see that (1/2) * (x + y, x) = (x, y), proving that (Ax, Ay) = (x, y) for any vectors x and y in R2.
Therefore, the given matrix A is an orthogonal matrix.