3
$\begingroup$

1) is there a math object or number (any type; quaternion or its further generalization, for example) that becomes zero when some operation is applied to the object using itself? The object must not be zero. also, when does such object not become zero when operated on it by another object?

2) is there an object that becomes zero when operated(multiplied) by another object?

3) is there any math object or matrix that satisfies the following property (I ask for the object that satisfies both constraints):

a)$ab+ba=0$ where a and b are math objects or matrices

b) $A^2=B^2$

for all objects I assume nonzero.

  • 1
    For (3a) this is a property called anticommutativity. The cross product of vectors is one example.2012-10-26
  • 0
    @Arthur edited.2012-10-26
  • 4
    See [zero divisor](http://en.wikipedia.org/wiki/Zero_divisor), [dual number](http://en.wikipedia.org/wiki/Dual_number)2012-10-26
  • 0
    @ttyy cross product of vectors still apply. If $ab + ba = 0$ for any $a, b$ from some specified algebraic structure, we have $aa + aa = 0$, thus as long as $2\neq 0$, any square is $0$ and therefore equal to any other square. So (3a) implies (3b) in most cases.2012-10-26
  • 0
    I'm fiddling with the question, whether the inverse of a certain infinite sized vandermondematrix is the Null-matrix. I could prove Nullity for some entries (assuming I've the correct formulae for the inverses of the LDU-factors). If you like you might look at http://go.helms-net.de/math/divers/InverseNullmatrix.pdf and possibly help me proceed (outside of this site)...2012-10-26

1 Answers 1