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What do we call a matrix whose columns are orthogonal, such as $\begin{bmatrix}3 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 0\end{bmatrix}$?
Is there a special name for it? I tried searching to no avail.

user541686
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    perhaps http://en.wikipedia.org/wiki/Orthogonal_matrix – vadim123 May 08 '13 at 23:19
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    That is not an orthogonal matrix. – Ross B. May 08 '13 at 23:21
  • @vadim123: Re-read the first sentence of the page you linked to, it's (sadly) not the same thing as you'd expect. – user541686 May 08 '13 at 23:22
  • An orthogonal matrix has orthonormal columns. –  May 08 '13 at 23:22
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    A rather unfortunate choice in nomenclature. – Ross B. May 08 '13 at 23:23
  • It may not be an orthogonal matrix, but if the columns are considered vectors, all three vectors are orthogonal to each other, verify that each dot product is zero. Now having a set of orthogonal vectors has applications in more than one math branch. Is this a bit helpful, Mehrdad? – imranfat May 08 '13 at 23:24
  • @vadim123 You just made me realize that orthogonal matrices should be called orthonormal matrices, somehow. It is funny that in the complex case, that's the other half that is kept: unitary matrices. – Julien May 08 '13 at 23:48

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There is no specific name for it, though such matrices are sometimes called scaled orthonormal/orthogonal matrices. An orthonormal/orthogonal matrix, say $Q$, is a square matrix that satisfies $$Q^*Q = I$$ where $Q^*$ is the conjugate-transpose. The matrices you consider are of the form $$DQ \text{ or } QD$$ where $D$ is a diagonal matrix.

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    By searching in google I don't find the term "scaled orthonormal/orthogonal matrices"!! –  May 08 '13 at 23:31
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    @SamiBenRomdhane You will find the term. In fact, if you google "scaled orthonormal/orthogonal matrices", you will find that the first post is this. –  May 08 '13 at 23:33
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Well, if the columns are orthonormal (i.e. norm 1), then the matrix is orthogonal, and has many beautiful properties.

If not, see Name for matrices with orthogonal (not necessarily orthonormal) rows.

I suppose the right way to think about it is that this matrix maps the standard basis vectors to an orthogonal basis. So it isnt quite preserving distance and angles, but its not too bad.

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Dhruv Ranganathan
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