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I have written something that says along the line

"Let matrix $A\in\mathbb{R}^{n \times n}$ be "earth-shattering", and assume that $A$ is not identically zero"

What I want to express is that $A$ is not the zero matrix.

Q1: Should I simply say that $A$ is not zero?

I am not trained on proper math language, but in the back of my mind "identically zero" means "zero everywhere", which seems most appropriate when we are discussing functions.

Q2: What is the proper way to use "identically zero"? What can and what can't be "identically zero"?

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    $\begingroup$ I think "identically zero" makes sense whenever the entries of A are functions. $\endgroup$ – Davey Dec 24 '16 at 4:43
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    $\begingroup$ I would just say "$A$ is not zero" or "$A$ is not the zero matrix". $\endgroup$ – littleO Dec 24 '16 at 4:49
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You are correct - when we say something is "identically" some value, generally we are talking about a function taking on that value everywhere. So it doesn't make much sense to say that a matrix is "identically zero". Though it is possible to think of a matrix as a linear transformation, in which case you can say that the linear transformation associated with the matrix is identically zero. This is true if and only if $A$ is the zero matrix.

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The phrase "identically zero" is generally used when we need to distinguish between a function having a zero at some point and a function being the zero function. Either might be written $f(x)=0$, for instance, so it helps to have a way to distinguish the two cases.

In the case of a matrix I'd just say the matrix is nonzero or whatever.

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Usually this would entail that there's some sort of dependence going on, that some of the entries of $A$ are functions, and while for certain inputs, it's alright that $A$ zeroes out, you are trying to avoid just having $A_{ij}(x) := 0$

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