Thank you for your response.
Background: I am just learning the notion of tensor rank. My real effort was understanding a certain article where
2x2x2 and 4x4x4 etc, arrays are involved. I was warming up, so to speak, and ran into some problems.
I do know the definition of tensor rank I believe. It is the minimum number of "diads" in the decomposition (in the case of a matrix), and the minimum number of "triads"
used in the decomposition for a 2x2x2 array.
So if @ denotes the tensor rank, the matrix I have posted shoud have tensor rank 2 since it can be written as:
(1,0) @ (1,1) + (0,1) @ (0,1)
Here is my question:
I will write a 2x2x2 array below; by first writing the front face as a matrix and the back face as a matrix.
Front |-1 0| Back |0 1|
|0 1| |1 0|
In this article it is asserted that the rank if this 2x2x2 array is 3, BUT it 2 (not 3) if the entries are considered to be Complex numbers instead of Real numbers.
I was trying to verify this (I was not able to).
If the assertion is correct, then, while the rank of a matrix does not depend on the base field of the entries, it does depend on the filed for a 2x2x2 array.
Any help is appreciated in verifying that the tensor rank is 2 for this array when considered as Complex entries. I thank you for your time
