I remember that every embedding is injective, and every projection is surjective. For square matrices, they're both embedding and projection, which means they're bijective, so they should be invertible. But obviously, not every square matrix is invertible. I don't which part is wrong in my logic. Not every projection is surjective? Or square matrices are not embedding or projection?
paradox of square matrix
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$\begingroup$
linear-algebra
matrices
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7Where do you get from that a square matrix is both "embedding and projection"? Both claims are false (and, btw, what you probably meant is that square matrices are, when considered as maps, both injective (or $1-1$) and suprajective (or onto), which is also false, of course) – 2012-06-28
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2@DonAntonio suprajective = surjective – 2012-06-28
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0Thanks @Marvis. It is becoming increasingly simpler (for me, of course) to use the terms from spanish into english...:) Oh, well: you all will learn. – 2012-06-28
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0@DonAntonio The tricky thing with "surjective" is that we got it from French, I think from the Bourbakistas. If it had been a more typical English word it would have been "superjective". – 2012-06-28
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1@Mark, typical English words come from French. Surplus, surcharge, surmount, surtax, .... – 2012-06-28
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0@Gerry I have not counted carefully, but I think "super-" words outnumber "sur-" words by about five to one. – 2012-06-28
1 Answers
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Consider the simplest example:
$$M: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$$
defined by the matrix $M = \begin{bmatrix} 0 & 0 \\ 0 & 0\end{bmatrix}$ is neither injective nor surjective. So square matrices needn't be injective nor surjective.