Show orthogonal upper triangular matrix is diagonal

Question

How do I go about showing that an upper triangular matrix with orthogonal columns has to be a diagonal matrix?

I know that the property $M^\text{T}M = I$ should be used but I'm not sure how.

Why not look at an off-diagonal component of $M^{\mathrm{T}}M = I$? — user26872, Apr 08 '12 at 23:08

score 10 · Accepted Answer · answered Apr 09 '12 at 00:11

Hint: If $M$ is any invertible upper triangular matrix, then $M^{-1}$ is also upper triangular (provable via, say, Cramer's rule), while $M^t$ is lower triangular (provable via, say, direct inspection). If $M$ is orthgonal, what's the relationship between $M^t$ and $M^{-1}$?

score 5 · Answer 2 · answered Apr 08 '12 at 23:14

5

We write the matrix as $[v^1...v^n]$. It is orthogonal, so we have $v^1\cdot v^n=0$, and that implies $v^n_1=0$. Then we proceed by induction.

answered Apr 08 '12 at 23:14

Juan

51

Show orthogonal upper triangular matrix is diagonal

2 Answers2