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1. Let's see what is matrix dot product. See https://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29 SMath defines n'th power of a matrix as dot product of n such matrices. By definition of a dot product, only square matrices may be raised to a power.

M 123522 mat :

M M * 712183122 mat

M 11 el M 11 el * M 12 el M 21 el * + M 11 el M 12 el * M 12 el M 22 el * + M 21 el M 11 el * M 22 el M 21 el * + M 21 el M 12 el * M 22 el M 22 el * + 22 mat 712183122 mat

M 2^712183122 mat

row 1 M rows range col 1 M cols range MM row col el M row k el M k col el * k 1 M cols sum : for for

MM 712183122 mat

13524623 mat 2^#

lastError Number of rows and/or columns of matrices or vectors do not match.

So, n'th power of a square matrix is a square matrix itself, where each element isn't equal to original matrix' corresponding element raised to nth power.

2. What is vector dot product (scalar multiplication). See https://en.wikipedia.org/wiki/Dot_product SMath defines n'th power of a vector as sequentially taking dot product of n such vectors.

2321 mat 2321 mat * 13

22 * 33 * 21 mat sum 13

132321 mat * 263921 mat

2321 mat 2321 mat * 2321 mat * 263921 mat

2321 mat 2^13

2321 mat 3^263921 mat

So, n'th power of a vector is a scalar, if n is even, or a vector, if n is odd. Its elements are not equal to original vector's corresponding element raised to nth power.

Now let's define a formula that raises a variable to a power

f x n^:

What is the kind of result of the function? Nobody knows, until its arguments are defined. If x happens to be scalar, the answer is a scalar. If x is a square matrix, f is a matrix, too. If x is a vector, then f is either vector or a scalar, dependent on n. If M is non-square matrix, result is error.

But what if I want to modify the dot behaviour, so that it operates not on matrices/vectors, but on their elements, so if a matrix/vector is supplied, then the same-dimentional matrix/vector is always returned with original input's elements raised to nth power? Use vectorize() for that. This function tells SMath that it shall not pass matrices/vectors to functions, but instead only matrix/vector elements. So, vectorize(M^n) is equal to "take M; for each element in M, do element^n; put each result to corresponding place in resulting structure". If M happens to be a scalar, then vectorize() will do nothing: it doesn't modify its scalar arguments, like n in previous case. So, M is safely any of scalar; vector; any matrix. And you get exactly what you want.

M f1 M n^vectorize :

n 3 :

123422 mat f1 18276422 mat

12345623 mat f1 18276412521623 mat

1421 mat f1 16421 mat

3 f1 27

I don't know MathCAD. I don't know how does it defines dot product of matrices/vectors. It may define them as element-wize power right away, and do just what you want; or it may define them as in SMath, but use some "clever" techniques to guess what is the most probable user's intention in this specific case. In either way, it's MathCAD's specifics; and that modus operandi is questionable, as any other decision. SMath doesn't need to be bit-to-bit MathCAD replacement; it has some properties that are absent in MathCAD. And personally I like that SMath makes my intention clear by using vectorize() instead of some hidden heuristics that *MAY* happen to co-incide with someone's expectations, but may be opposite to someone's other.

If you really need to eliminate SMath's vectorize() in the case of f:=x^n, then you need to use programming (tests and loops) like that:

M f2 MM M rows M cols matrix : row 1 M rows range col 1 M cols range MM row col el M row col el (n^: for for MM 31 line Error? M must be scalar? M n^21 line try 11 line :

n 3 :

123422 mat f2 18276422 mat

12345623 mat f2 18276412521623 mat

1421 mat f2 16421 mat

3 f2 27

But I'm afraid that's not good.