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The columns of matrix $\matrix M$ are orthogonal to each other

What does this statement mean? Aren't columns parallel to each other? How can columns be orthogonal- aren't columns parallel to each other vertically by definition?

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It doesn't mean that the columns are literally orthogonal on the page. It means that two columns, when considered as separate vectors, represent orthogonal vectors. –  MJD May 22 '12 at 13:48
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Even if not meant as one, this makes an excellent joke. –  Did May 22 '12 at 13:49

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Two vectors $x$ and $y$ are said to be orthogonal if

$$x\cdot y = 0$$

where $\cdot$ is the standard dot product on vectors:

$$x\cdot y = x_1y_1 + \cdots + x_n y_n$$

When someone says that the columns of a matrix are orthogonal, they mean that if you consider each of the columns to be a vector, those vectors are all orthogonal to each other.

In a little more detail, an $n\times n$ matrix can be viewed as $n$ vectors stacked vertically next to each other. Call the vectors $v_1$, ..., $v_n$. Then if the columns are orthogonal, we have

$$v_i \cdot v_j = 0$$

for every $i$ and $j$.

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The columns of an orthogonal matrix are also assumed to have norm one, so an orthogonal matrix is actually "orthonormal" from this perspective. –  t.b. May 22 '12 at 13:49
    
I wondered if the author might be drawing a distinction between "the columns of the matrix are orthogonal" and "the matrix is orthogonal" since you can certainly have a matrix whose columns are orthogonal which is not an orthogonal matrix (although this would be a weird thing to think about...) –  Chris Taylor May 22 '12 at 13:52
    
Maybe you should say a word or two why the definition of orthogonal vectors makes sense... –  Simon Markett May 22 '12 at 14:00
    
Not a bad idea. –  Chris Taylor May 22 '12 at 14:06
    
The book then goes on to state that since the columns are orthogonal, they can be thought of as the axis of an alternate coordinate system (Fourier basis). If the columns aren't literally right angled, why do they have a relation with the right angled axis of coordinate systems? –  user26649 May 22 '12 at 14:40

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