Matrix Reference Manual_ Matrix Calculus.pdf

12/30/12 Matrix Ref erence Manual: Matrix Calculus
Matrix Calculus
Go to: Introduction, Notation, Index
Contents of Calculus Section
Notation
Differentials of Linear, Quadratic and Cubic Products
Differentials of Inverses, Trace and Determinant
Hessian matrices
Notation
j is the square root of -1
XR and XI are the real and imaginary parts of X = XR + jXI
XC is plex conjugate of X
X: denotes the long column vector formed by concatenating the columns of X (see vectorization).
A ¤ B = KRON(A,B), the kroneker product
A • B the Hadamard or elementwise product
matrices and vectors A, B, C do not depend on X
Derivatives
In the main part of this page we express results in terms of differentials rather than derivatives for two reasons: they avoid notational disagreements and
they cope easily with plex case. In most cases however, the differentials have been written in the form dY: = dY/dX dX: so that the
corresponding derivative may be easily extracted.
Derivatives with respect to a real matrix
If X is p#q and Y is m#n, then dY: = dY/dX dX: where the derivative dY/dX is a large mn#pq matrix. If X and/or Y are column vectors or scalars,
.tuhke/h pv/setcaftfo/drmizba/mtiaotrnix / l: has no effect and may be omitted. dY/dX is also called the Jacobian Matrix of Y: with respect to X: and 1/7
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