Kronecker Products and Matrix Calculus: With Applications by Alexander Graham

By Alexander Graham

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22), we can write o(UV) as one would expect. 25) p=1 s. 6 Let X = lX,sJ be a non-singular matrix. 24) to differentiate yy-\ = I, we obtain ay-I ay -y-I+ y _ = 0, ax,s ax,s hence ay-I -y-y -. 20» . 15) which is valid . 20) and conversely. 20) we replace A by A', B by B' and Ers by Eii (careful,E,s and Ell may be of different orders). 20) were derived for constant matrices A and B, the above transformation is indepcndcnt of the status of the matrices and Is valid even when A and n arc functions of X.

We consider a matrix H(n X n) whose eigenvalues are the desired values fq, Al' ... 30) whereP= e'@B,k=vecK and q = vec Q. Notice that P is of order (n 2 X mr) and k and q are column vectors of order mr and n2 respectively. 30) is overdetermined unless of course m = n = r, in which case can be solved in the usual manner - assuming a solution does exist! In general, to solve the system for k we must consider the subsystem of linearly independent equations, the remaining equations being linearly dependent Sec.

We will consider a general case, say we have a matrix Y = [Yi/] whose components are functions of a matrix X = [Xii]' that is YI/ = Ii/(x) where x = [xu xu··· xmnJ'. 4J The Derivative of Scalar Functions of a Matrix 57 We will determine which will allow us to build up the matrix a/YI ax Using the chain rule we can write olYI - - == where Yi/15 the cofactor of the elementYl/ln IY!. , ... olYI aYi! 8) Although we have achieved our objective in determining the above formula, it can be written in an alternate and useful form.

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