Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm told that zero power of square matrix is an identity matrix of appropriate size. How it is with a non-square matrix?

share|improve this question
add comment

2 Answers

up vote 4 down vote accepted

Exponentiation of (square) matrices is a recursive function, which we define so that $A^0 =I$ and $A^1 = AA^0 = A^0A$, $A^2 = AA^1= A^1A$, etc.

Exponentiation of non-square matrices is not well-defined, for the simple reason that $A^2 =AA$ (and higher products of $A$ with itself) is not a valid matrix product. So, the short answer is that because we don't define exponentiation at all for non-square matrices, even $A^0$ and $A^1$ are undefined in the non-square case.

Suppose you insisted on having a definition of $A^0$ and $A^1$ for m×n matrices for mn, so that $A^1=A$ and also $A^0$ was some matrix which gave you back $A$ after left- or right-multiplication by $A$. But that's absurd, because whatever you tried to define $A^0$ as, only one of those products would be defined. So we don't really have any good, unique choice for what $A^0$ should be that would also give us a reasonable definition of $A^1$ that we would expect. Lacking any good way to define $A^0$, we leave both $A^0$ and $A^1$ undefined.

share|improve this answer
add comment

The notation of "power" makes no sense for non-square rectangular matrices since you can't multiply them by themselves. You can define a zero power of a non-square matrix to be what you like, but it will probably be of little use.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.