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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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    Even if not meant as one, this makes an excellent joke.2012-05-22

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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 when looked at as vectors are indeed "literally" right angled. Hence, it makes for an orthogonal basis.2012-05-29