Transposition of a matrix

  1. Why is the transpose of a matrix important?
    To find the inverse by cofactors we need the transpose but I would never find the inverse of a matrix by using cofactors.
     
  2. jcsd
  3. Fredrik

    Fredrik 10,543
    Staff Emeritus
    Science Advisor
    Gold Member

    I think the main reason why the transpose is useful is that the standard inner product on the vector space of nĂ—1 matrices is [itex]\langle x,y\rangle=x^Ty[/itex]. This implies that a rotation R must satisfy [itex]R^TR=I[/itex].

    I think that cofactor stuff is sometimes useful in proofs, but you're right that if you just want to find the inverse of a given matrix, there are better ways to do it.
     
  4. There are of course many ways to invert a matrix but thie is not the only use for the transpose.

    Systems of linear equations can be reformulated into matrix systems by looking at the equation xAx^{T} = b where x is a n x 1 column vector with entries {x_{1},...,x_{n}} and Z is a square matrix n x n with entries (for real valued equations, say) in /mathbb{R}. The matrix b is then also an $n x 1$ column matrix of numbers in /mathbb{R} too.
     
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