# Spark vs Rank of a matrix

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I understand the rank of a matrix to be the maximum number of linearly independent rows (or equivalently columns) in a matrix.

The spark seems to be the minimum number of columns which are linearly dependent.

I'm missing some subtlety because these seem to me to be the same thing. Does anyone know any good resources on the distinction, i could only find the wikipedia page and that is a bit sparse

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Consider an invertible 2 by 2 matrix. The rank would be 2, while the spark would be infinite. Or for the zero matrix, the rank is 0, while the spark is 1.

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For the first one an infinite spark implies a full rank anyway. Not sure why the spark is 1 iff a matrix has a zero column (is it just defined as such?).

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Because the spark is the minimum number of linearly dependent columns. If it has a zero column, then that column is linearly dependent by itself - so the spark is as small as it can be, 1.

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I don't get why a full rank matrix has a spark infinite. Shouldn't it be zero?

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The statement "the spark of a matrix is zero" expands to mean "There is a set of columns of size zero that is linearly dependent." Which isn't true.

Spark of a full rank matrix is something of a convention. Spark increases as linear dependence decreases - and a full rank matrix is maximally linearly independent, so you want the spark to be large, not small. Choosing it to be infinity is likely the best convention.

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