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Outer products & positive (semi-) definiteness

  1. Oct 30, 2015 #1
    Let ##u,v## be vectors in the same Euclidean space, and define the symmetric matrix ##M = uv'+vu'##, the sum of their two outer products.

    I'm interested in whether or not ##M## is positive (semi)definite.

    Does anybody know of any equivalent conditions that I might phrase "directly" in terms of the vectors ##u,v##?
     
  2. jcsd
  3. Oct 30, 2015 #2

    andrewkirk

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    An equivalent condition is ##(x\cdot u)(x\cdot v)\geq 0## for all vectors ##x##.
     
  4. Oct 30, 2015 #3

    fzero

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    Let ##z## be an arbitrary vector, then we are interested in the properties of
    $$z^\dagger M z = \langle z, u \rangle \langle v, z \rangle + \langle z, v \rangle \langle u, z \rangle .$$
    Assume that ##v\neq u##, then we can write
    $$ z = a u + b v + z_\perp,~~~\langle z_\perp, u \rangle = \langle z_\perp, v \rangle =0.$$
    Then, with ## A = (a~~b)^T##, we have
    $$z^\dagger M z = A^\dagger Q A,$$
    where ##Q## is a Hermitian 2x2 form determined in terms of the inner products of ##u## and ##v##. Positivity properties of ##M## are reduced to those of ##Q## for which it is simple to compute the eigenvalues.
     
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