How can I generally solve equations of the form A w = ( x y...

Faith Welch

Faith Welch

Answered

2022-07-19

How can I generally solve equations of the form A w = ( x y z ) × w for the matrix A, where w can be any vector? I recognize that you could just set w to a vector with simple values, such as ( 1 2 1 ) , but doing so still isn't helpful. Also, x, y, and z are entirely independent variables.

Answer & Explanation

minotaurafe

minotaurafe

Expert

2022-07-20Added 22 answers

OK, let's put it other way as w × v = A w . We can write the the cross product as vector-matrix multiplication:
w × v = [ w ] × v = [ 0 w 3 w 2 w 3 0 w 1 w 2 w 1 0 ] v .
So you can write your equation as a system of linear equations
[ w ] × v = A w .
Matrix [ w ] × has rank 2 and its nullspace is spanned by [ w 1 , w 2 , w 3 ]
Now depending on whether you assume w 2 0 or w 3 0, you can transform this system and find a particular solution. However, this solution can be found only if w , A w = 0. In particular, this implies that A = A

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?