Show that matrix A 13 23 23 23 13 23 23 23 13 is orthogonal

Show that matrix A = [1/3 -2/3 2/3 2/3 -1/3 -2/3 2/3 2/3 1/3] is orthogonal and find A^-1

Solution

We know that an nxn orthogonal matrix or is a square matrix whose columns and rows are orthogonal unit vectors. Also, for an orthogonal matrix A, we have A^TA= AA^T = I_n so that A^T = A^-1.

Here, let the column vectors of the given matrix A be denoted by v₁,v₂ and v₃ i. e. let v = (1/3, 2/3,2/3)^T, v₂ = (-2/3, -1/3,2/3)^T, and v₃ = (2/3, -2/3,1/3)^T. Then v₁.v₂ = (-2/9)-(2/9)+(4/9) = 0, v₂.v₃ = (-4/9)+ (2/9) + (2/9) = 0 and v₃.v₁ = (2/9)-(4/9)+(2/9) = 0. Hence, v₁,v₂ and v₃ are orthogonal to one another. Further, ||v₁|| = [ (1/3)²+(2/3)²+ (2/3)²] = (1/9 +4/9+ 4/9) = 1 = 1, ||v₂|| = [ (-2/3)²+(-1/3)²+ (2/3)²] = (4/9 +1/9+ 4/9) = 1 = 1   and ||v₃|| = [ (2/3)²+(-2/3)²+ (1/3)²] =(4/9 +4/9+ 1/9) = 1 = 1. Hence, v₁,v₂ and v₃ are orthogonal unit vectors so that A is an orthogonal matrix. Then A^-1 = A^T =

1/3

2/3

2/3

-2/3

-1/3

2/3

2/3

-2/3

1/3

1/3	2/3	2/3
-2/3	-1/3	2/3
2/3	-2/3	1/3