Given matrices A,B,C and column vector v , what is the derivative of ⟨ A diag ( B v ) , C ⟩ with respect to B ? diag(⋅) is a diagonal matrix with the argument as the diagonal and the brackets signify the inner product.I am trying to find the solution, but am hung up on differentiating the diagonal matrix.

Question

Given matrices A,B,C and column vector       v  , what is the derivative of   ⟨      A    diag  (      B        v    )  ,      C    ⟩ with respect to       B  ? diag(⋅) is a diagonal matrix with the argument as the diagonal and the brackets signify the inner product.I am trying to find the solution, but am hung up on differentiating the diagonal matrix.

megagoalai · Accepted Answer

The derivative can always be computed by going back to the original limit definition. For a vector-valued function   f  :  X  →  Y, its derivative at a point x, denoted   D  f  (  x  ), if it exists, is given by  [  D  f  (  x  )  ]  z  =      lim          h      →      0            h          −      1        (  f  (  x  +  h  z  )  −  f  (  x  )  )  .In other words, if the true (Frechet) derivative exists, then you can compute it by computing directional (Gateaux) derivatives.In your case,   f  (  X  )  =  ⟨  A      d    i    a    g    (  X  v  )  ,  C  ⟩, where X is the variable (B in your original question). Then perhaps you can verify that  [  D  f  (  X  )  ]  Z  =  ⟨  A      d    i    a    g    (  Z  v  )  ,  C  ⟩

Mattie Monroe · Accepted Answer

Let rewrite the cost function  ϕ  =      A        d    i    a    g    (      B    v    )  :      C    =      d    i    a    g    (      B    v    )  :            A        T        C    =      B    v    :      d    i    a    g    (            A        T        C    )  =      d    i    a    g    (            A        T        C    )            v      T        :      B  I used the colon operator for the Frobenius inner product. Also diag(X) extracts the diagonal element of X into a vector. The derivative is            ∂      ϕ              ∂              B              =      d    i    a    g    (            A        T        C    )            v      T

Given matrices A,B,C and column vector v, what is the derivative of ⟨Adiag(Bv),C⟩with respect to B?

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