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I am pretty sure the definition of orthonormal matrix is a matrix whos columns contain vectors that are orthogonal to each other and all of length 1.

In that case I cant understand 17:04 of this lecture:https://www.youtube.com/watch?v=uNsCkP9mgRk&t=974s.

Professor Gilbert strang writes that $Q =\frac{1}{3}\begin{pmatrix}1&-2& 2\\2&-1&-2\\2&2&1\end{pmatrix}$ and says the columns are orthonormal to each other. I dont see how since the dot product of the vectors isn't 0. Is this a mistake in the lecture or am I missing something.

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  • $\begingroup$ Can you please tell use which two columns are not orthogonal? $\endgroup$ – José Carlos Santos May 12 at 8:33
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    $\begingroup$ Vectors are orthogonal if their dot product IS zero. The columns of this matrix are pairwise orthogonal, because their dot products are all zero. Also, all of their lengths are 1. We call such a matrix orthogonal, and not orthonormal, btw. $\endgroup$ – Ivo Terek May 12 at 8:34
  • $\begingroup$ Wait if i have a vector $[1\ 2\ 2]^T$ and dot it with $[-2 -1\ 2]^T$ it is not 0. So how come we can say this is an orthogonal matrix? $\endgroup$ – Vishal Jain May 12 at 8:38
  • $\begingroup$ I would say that $1\times(-2)+2\times(-1)+2\times2$ is $0$. What do you think it is? $\endgroup$ – José Carlos Santos May 12 at 8:39
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    $\begingroup$ ... I can't believe I've been doing the dot product wrong.. thanks $\endgroup$ – Vishal Jain May 12 at 8:41
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Your matrix is orthogonal:

  • every column has norm $1$, since the norm of each column is $\sqrt{\frac49+\frac49+\frac19}=1$;
  • the dot product of every two distinct columns is $0$; for instance, the dot product of the first two columns is$$\frac19\bigl(1\times(-2)+2\times(-1)+2\times2\bigr)=0.$$
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Firstly, the columns each have length 1. For example, the first column which is $(1/3,2/3,2/3)^T$ has length

$$\sqrt{(1/3)^2 + (2/3)^2 + (2/3)^2} = \sqrt{1/9 + 4/9 + 4/9} = 1.$$

The second and third columns are basically the same. As for being normal, the dot products of any two of the columns is equal to zero. For example,

$$(1,2,2)^T \cdot (-2,-1,2)^T = 1(-2) + 2(-1) + 2(2) = -2 - 2 + 4 = 0. $$

It should be easy to check that the second and third and first and third columns are orthogonal to each other.

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  • $\begingroup$ What has this to do with the question? The OP is asking about the dot product of the columns, not about their norms. $\endgroup$ – José Carlos Santos May 12 at 8:38
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The definition of an orthonormal matrix is for

$$Q^T\cdot Q = Id = Q \cdot Q^T$$

to hold.

Calculating this, we get

$$\frac{1}{9}\begin{bmatrix}1 & 2 & 2 \\-2 & -1 & 2 \\ 2 & -2 & 1\end{bmatrix}\cdot\begin{bmatrix}1 & -2 & 2 \\ 2 & -1 & -2 \\ 2 & 2 & 1\end{bmatrix} = \frac{1}{9}\cdot 9 \cdot Id = Id$$

so the matrix is by definition orthonormal.

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