What 2 by 2 matrix R rotates every vector through 45 degree ? The vector (1,0) goes to (√2/2, √2/2). The vector (0,1) goes to (-√2/2,√/2).

== Solution ==

Consider (x,y) as:

  x = k * cos(α)
  y = k * sin(α)

where k is the distance from (0,0) and α the angle from x-axis.

Rotating it through 45 degree:

  x = k * cos(α + π/4)
  y = k * sin(α + π/4)

By trigonometric identities,

  x = k * [cos(α) * cos(π/4) - sin(α) * sin(π/4)]
    = k * [cos(α) - sin(α)] / √2
    = (x - y) / √2

  y = k * [sin(α) * cos(π/4) + cos(α) * sin(π/4)]
    = k * [sin(α) + cos(α)] / √2
    = (x + y) / √2

What matrix would act on (x,y) to yield (x-y, x+y) ?

   |1 -1||x|   |x-y|
   |    || | = |   |
   |1  1||y|   |x+y|

So the answer is:

   |1/√2 -1/√2|
   |1/√2  1/√2|

# posted by rot13(Unafba Pune) @ 10:43 AM