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What is the more generalized term? Why is MATLAB named matrix laboratory, then?
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The arrays in Python are ndarray objects. The matrix objects are strictly 2-dimensional whereas the ndarray objects can be multi-dimensional.
Matrix operations follow the rules of linear algebra. By contrast, array operations execute element by element operations and support multidimensional arrays. The period character ( . ) distinguishes the array operations from the matrix operations.
A matrix is a 2D array with which follows the rules for linear algebra. It is, therefore, a subset of more general arrays which may be of higher dimension or not necessarily follow matrix algebra rules.
A matrix is a two-dimensional vector (fixed size, all cell types the same). An array is a vector with one or more dimensions. So, an array with one dimension is (almost) the same as a vector. An array with two dimensions is (almost) the same as a matrix.
A matrix is a practical way to represent a linear transformation from a space of dimension n to a space of dimension m in the form of a nxm array of scalar values.
It is also very practical to perform linear algebra operation in a very systematic way that can be implemented on a computer. For instance if matrix A represents the linear transformation f and matrix B the linear transformation g, then the composition f o g writes as A*B where * denotes matrix multiplication. Matlab has also a lot of routines related to matrix operations (i.e. linear algebra operations) like det, pinv, svd etc...
As you can still see nowadays in Matlab, operators like *, / are strongly tied to matrix operations and thus strongly tied to linear algebra operations, which I think was the original goal of matlab in its early elaboration, hence its name (surely quite speculative but guess not so far from reality).
To perform element-wise operations on n-dimensional data sets, you have to write .*, or ./. denoting you are now performing array operations.
I would not say array operations encompass matrix operations, they are different. The later ones relate to linear algebra, while the other ones just relate to a practical way to operate on large sets of data. These data are not limited to be numbers, they are just n-dimensional data sets of whatever (string, numbers, cells, etc...).
Matlab also has a very synthetic syntax to perform array operations on sub-blocks (i.e. linear/logical subscripts) that makes it very easy to reorganize data sets in just one line of code before applying subsequent matrix or array operations.
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