Condition number of empty matrix

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Robert van de Geijn 2022-6-20

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评论： Robert van de Geijn 2022-6-23

if is a 0 x 0 matrix, then cond( A ) returns 0. However, we know that the condition number is always >= 1. Also, one could argue that the empty matrix is an identity matrix, and hence its condition number should equal 1.

Any particular reason why matlab made this choice?

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Robert van de Geijn 2022-6-22

I don't think you should think about it that way.

The condition number is the maximum by which a vector is stretched divided by the minimum that a vector is stretched. The only vector that exists is the 0 x 1 vector. When you apply a 0 x 0 matrix to it, you get that same vector back, regardless. Hence the condition number, if defined at all, should equal 1.

Robert van de Geijn 2022-6-22

Let me now complicate matters even more with some more musings...

However, if anything, x being a 0 x 1 vector (a vector with no elements), its length (2-norm) must be 0, and hence it is the zero vector. Thus, the 2-norm of a 0 x 0 matrix cannot be defined...

If the 2-norm cannot be defined, then the condition number cannot be defined...

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Answer 1

Christine Tobler 2022-6-23

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The case of a 0-by-0 matrix doesn't have any very useful definition, as you note correctly in the comments above.

MATLAB does what it does because it computes any p-condition number using the formula:

norm(A, p) * norm(inv(A), p)

and of course the norm of [] is 0, as is the norm of the inverse of [].

A legitimate question could be if the norm of a [] matrix should be 0, or if it should be NaN since this matrix can't be mulitplied with a vector that has norm 1. But in practical terms, I think it's more useful to define this norm as being 0 than returning NaN.

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John D'Errico 2022-6-23

编辑：John D'Errico 2022-6-23

在 MATLAB Online 中打开

Just thinking, I'll play the devil's advocate here, arguing it should return NaN.

What does it mean, when a result comes back as NaN? NaN means to me that a computation could have taken on any of multiple values, and probably infinitely many values. For example, what is sin(inf)? If we try to take the limit of sin(X), as x approaxhes infinity, clearly it could be any value from the closed interval [-1,1]. And so we assign a NaN to that result.

Similarly, what is inf/inf? Again, we can answer that only in context of a limit. But what matters is how fast you approach that limit in the numerator and the denominator. For example, we could ask for the limit of (x/x^2), as x approaches infinity. Clearly that would be 0. But suppose we tried to take the limit of 17*x/x, as x approaches inf? Now we could argue that inf/inf should be the number 17. The point is, inf/inf can only be returned as the not-a-number result of NaN, because we could write it as any limits we choose, and we can get infinitely many different results. 0/0 is the same thing, even though there are passionate pleas to be found online for 0/0 to have some specific value, usually 1.

The point being, a NaN should be returned when a result could arguably be anything.

Is that a valid argument to apply to cond([])?

Christine makes a valid argument that it should be an empty result, since norm([]) is empty and inv([]) also is empty. Each of these follow the MATLAB convention to return an empty output for empty input. And of course,

[]/[]
ans =

     []

So perhaps what we need to argue is if norm([]) should be zero? Should inv([]) be something other than []?

In both of those cases, it makes sense they should be []. Consider inv. inv(A) is a square matrix of the same size as A. So inv([]) CANNOT be a 1x1 scalar. It MUST be []. Similarly, what is norm([])? The 1-norm would have us take the sum of the absolute values of the sum of the elements of the input. That looks like [] to me, and I could make the same argument for norm([]). (Feel free to pick your favorite norm. Mine is Norm Abrams.)

Your honor, the devil's advocate would like to approach the bench, and ask to be removed from the case.

Paul 2022-6-23

编辑：Paul 2022-6-23

@Steven Lord, thanks for the link.

My assumption that TMW invented Matlab's [original] empty matrix implementation was based on the actual statement in the doc at that time (from that link's link to deBoor):

"'As far as we know, the literature on the algebra of empty matrices is itself empty. We're not sure we've done it correctly, or even consistently, but we have found the idea useful."

Perhaps I should have said: I always thought that the TMW implementation of Matlab's empty matrix was basically a TMW construct.

No matter, obviously. I just have an historical interest.

Robert van de Geijn 2022-6-23

As a purist, I consider a matrix to strictly be a convenient representation of a linear transformation. So, the first questions are

1) is R^0 (or C^0) a vector space?

2) Are there linear transformations that map R^0 to R^0?

3) How are these linear transformations represented as (0 x 0) matrices?

That establishes whether [] is even a matrix.

Let's say that the answers are

1) Yes, and its only element is the 0 x 1 vector, which then must be the zero vector, since the zero vector must be in the space.

2) Yes, only one: the linear transformation that maps the 0 x 1 to itself.

3) The matrix that represents it then is [] (the 0 x 0 matrix).

At this point we can start discussing the norm (let's stick to the 2-norm).

a) Since the 0 x 1 vector is the zero vector, its norm equals 0.

b) While the 2-norm for n > 0 is defined as max_x \neq 0 || A x ||, it is better to go back to thinking about linear transformations. The "size" (2-norm) of a linear transformation really measures by how much the linear transformation stretches the vector to which it is applied. One can argue that the linear transformation

a) maps the 0 x 1 vector to the zero vector and hence || A || = 0

b) maps the 0 x 1 vector to itself and hence A = I and || A || = 1

c) stetches the 0 x 1 vector by any nonnegative integer, since it is the zero vector.

The fact that all of these can be reasonably argued to me indicates that the condition number is ill-defined. Hence, I believe NaN or "not defined" are the only results in the running.

Now, it may be that since matlab is used as a tool for manipulating arrays in addition to matrices, it is convenient to define it differently. But then we should not really call it the condition number. It is merely the matlab operation that is denoted by cond().

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Condition number of empty matrix

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