A n × n matrix A is a linear transformation from the space of n × n matrices to itself. What is the n 2 × n 2 matrix corresponding to this transformation? If we knew the nullity and rank of A, how could we find the nullity and rank for this transformation?

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A   n  ×  n matrix   A is a linear transformation from the space of   n  ×  n matrices to itself. What is the       n    2    ×      n    2   matrix corresponding to this transformation? If we knew the nullity and rank of   A, how could we find the nullity and rank for this transformation?

knolsaadme · Accepted Answer

Let         R          n      ×      n       denote the set of   n by   n matrices with real entries. A matrix   A  ∈        R          n      ×      n       specifices a linear map       ϕ    A    :        R          n      ×      n        →        R          n      ×      n       through matrix matrix multipliciation, i.e  ∀  X  ∈        R          n      ×      n          :        ϕ    A    (  X  )  =  A  X  .By definition,       ϕ    A    (  X  )  =  A  X can be computed by computing   A      x    i   for each column vector       x    i   of   X . If we where to stack the columns of   X on top of each other, forming the gigantic vector       x    ~    ∈        R            n      2       given by       x    ~    =  (      x    1    T    ,      x    2    T    ,  ⋅  ,      x    n    T        )    T  and wanted to obtain a a similar representation of   Y  =  A  X, this could be achieved through the matrix vector multiplication      y    ~    =      A    ~        x    ~  where       A    ~    ∈        R                  n        2            ×              n        2             is obtained by using   n copies of   A as diagonal blocks in a gigantic matrix, i.e.      A    ~    =      (                            A                                                                      A                                                                      ⋱                                                                      A                      )  This form reveals that if   A has rank   r, then       A    ~   has rank   n  r.There is good mathematical notation for the objects that I have used here. Normally, one writes      x    ~    =  vec  (  X  )and      A    ~    =      I    n    ⊗  Awhere       I    n   is the   n by   n identity matrix and   ⊗ denotes the Kronecker product.

A n × n matrix A is a linear transformation from the space of

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