# direct products and direct sums for matrices and for vector spaces

1. I was wondering what relations and similarities are between direct product for matrices and direct product for vector spaces? Or do they just unfortunately and somehow misleadingly happen to have the same name?

Note that the direct product for matrices is also called Kronecker product or tensor product of matrices.

2. I was wondering if there is a similar thing for matrices just as direct product for vector spaces? Direct sum for matrices seems to correspond to direct sum for vector spaces, instead of direct product for vector spaces. So I guess direct sum for matrices is not the answer?

3. Thanks to Arturo for his comment:

You can connect direct sums of matrices with direct sums of vector spaces in the following sense: if $A$ is an $n×m$ matrix and $B$ is a $p×q$ matrix, then $A⊕B$ is the block diagonal matrix that has upper left block $A$ and bottom right block $B$. Interpreting $A$ as a map $F_m→F_n$ and $B$ as a map $F_q→F_p$, then $A⊕B$ is the corresponding map $F_m⊕F_q→F_n⊕F_p$.

Since direct sum and direct product of vector spaces share so much similarity, why is it direct sum instead of direct product of vector spaces that the direct sum of matrices correspond to?

Thanks!

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For finitely many factors, the direct product and the direct sum of vector spaces are isomorphic: they are a biproduct. Whether you call it direct sum or direct product is immaterial. –  Arturo Magidin Aug 15 '11 at 20:06
How about cases where there are not necessarily finitely many factors? They are not necessarily the same. –  Tim Aug 15 '11 at 20:10
But then how do you define a "direct sum of infinitely many matrices"? It's not a matrix any more. –  Arturo Magidin Aug 15 '11 at 20:13
It's misleading to call the tensor product a "direct product": it should just be called the tensor product. –  Qiaochu Yuan Aug 15 '11 at 20:17

There are three standard ways of combining a collection of vector spaces $V_i$, and in full generality they are all different:

• The direct sum $\bigoplus V_i$. Concretely it consists of the subspace of the direct product spanned by the image of the $V_i$. Abstractly it is the coproduct in the category of vector spaces.
• The direct product $\prod V_i$. Concretely it is the set-theoretic product. Abstractly it is the product in the category of vector spaces.
• The tensor product $\bigotimes V_i$. The concrete description seems messy for infinitely many factors. Abstractly it is neither the coproduct nor the product: instead it is universal with respect to multilinear maps out of the $V_i$.

The direct sum and direct product agree for finitely many factors but disagree in general; the tensor product almost never agrees with either.

So much for vector spaces; what about matrices? The universal properties of the direct sum and direct product can concisely be written as

$$\text{Hom}(\bigoplus V_i, W) \cong \prod \text{Hom}(V_i, W)$$

and

$$\text{Hom}(W, \prod V_i) \cong \prod \text{Hom}(W, V_i).$$

It follows that

$$\text{Hom}(\bigoplus V_i, \prod W_i) \cong \prod \text{Hom}(V_i, W_j).$$

So given a collection of maps $f_{ij} : V_i \to W_j$ we canonically get a map $f : \bigoplus V_i \to \prod W_i$. For finitely many factors we have $\prod W_i \cong \bigoplus W_i$, and so in this case I guess one could call $f$ the "direct sum" of the $f_{ij}$, although I think this is mildly misleading. I don't know a better term, though.

What I've described above is not the Kronecker product. The Kronecker product is a description in coordinates of an abstract way to combine a collection of maps $f_i : V_i \to W_i$ into a map $f : \bigotimes V_i \to \bigotimes W_i$, as follows: given a multilinear map $B$ from the $W_i$ to some vector space $U$, we can compose $B$ with each of the $f_i$ to get a multilinear map from the $V_i$ to $U$, and by the universal property we get the desired map $f$.

The Kronecker product is entirely defined in terms of the tensor product, and in particular makes no use of the direct product, so I think it is quite misleading to call it the "direct product."

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Thanks! (1) Could you elaborate how the universal properties of the direct sum and direct product can concisely be written as $Hom(⨁Vi,W)≅∏Hom(Vi,W)$ and $Hom(W,∏Vi)≅∏Hom(W,Vi)$? (2) In the part for kronecker product, when having a multilinear map from the Vi to U, how does the universal property lead to the desired map $f:⨂Vi→⨂Wi$? I only see $∏Vi→⨂Vi$ and $∏Wi→⨂Wi$? –  Tim Aug 15 '11 at 23:17
@Tim: 1) well, it is more or less a statement in formulas of the fact that "every collection of morphisms $f_i : V_i \to W$ uniquely factors through a morphism $f : \bigoplus V_i \to W$" and so forth. 2) Careful; the maps you write down aren't linear. First a multilinear map from the $W_i$ to $U$ gives a map $\bigotimes W_i \to U$ by the universal property, and the composition above gives a multilinear maps from the $V_i$ to $\bigotimes W_i$; then we invoke the universal property again. –  Qiaochu Yuan Aug 16 '11 at 2:43