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Notes on Matrix Calculus
Paul L. Fackler∗
North Carolina State University
September 27, 2005
Matrix calculus is concerned with rules for operating on functions of
matrices. For example, suppose that an m ×n matrix X is mapped into a
p×q matrix Y. We are interested in obtaining expressions for derivatives
such as
∂Yij ,
∂Xkl
for all i,j and k,l. The main difficulty here is keeping track of where things
are put. There is no reason to use subscripts; it is far better instead to use
a system for ordering the results using matrix operations.
Matrix calculus makes heavy use of the vec operator and Kronecker
products. The vec operator vectorizes a matrix by stacking its columns (it
is convention that column rather than row stacking is used). For example,
vectorizing the matrix
1 2
3 4
5 6
∗Paul L. Fackler is an Associate Professor in the Department of Agricultural and Re-
source Economics at North Carolina State University. These notes are copyrighted mate-
rial. They may be freely copied for individual use but should be appropriated referenced
in published work.
Mail: Department of Agricultural and Resource Economics
NCSU, Box 8109
Raleigh NC, 27695, USA
e-mail: paul fackler@ncsu.edu
c Web-site: http://www4.ncsu.edu/∼pfackler/
° 2005, Paul L. Fackler
1
produces
1
3
5
2
4
6
The Kronecker product of two matrices, A and B, where A is m×n and B
is p × q, is defined as
A B A B ... A B
11 12 1n
A B A B ... A B
A⊗B= 21 22 2n ,
... . . . . . . . . .
A B A B ... A B
m1 m2 mn
which is an mp × nq matrix. There is an important relationship between
the Kronecker product and the vec operator:
vec(AXB)=(B⊤⊗A)vec(X).
This relationship is extremely useful in deriving matrix calculus results.
Another matrix operator that will prove useful is one related to the vec
operator. Define the matrix Tm,n as the matrix that transforms vec(A) into
⊤
vec(A ):
⊤
T vec(A) = vec(A ).
m,n
Note the size of this matrix is mn × mn. T has a number of special
m,n
properties. The first is clear from its definition; if T is applied to the
m,n
vec of an m × n matrix and then T applied to the result, the original
n,m
vectorized matrix results:
T T vec(A) = vec(A).
n,m m,n
Thus
T T =I .
n,m m,n mn
The fact that
T =T−1
n,m m,n
follows directly. Perhaps less obvious is that
T =T ⊤
m,n n,m
2
(also combining these results means that T is an orthogonal matrix).
m,n
The matrix operator T is a permutation matrix, i.e., it is composed
m,n
of 0s and 1s, with a single 1 on each row and column. When premultiplying
another matrix, it simply rearranges the ordering of rows of that matrix
(postmultiplying by T rearranges columns).
m,n
The transpose matrix is also related to the Kronecker product. With A
and B defined as above,
B⊗A=T (A⊗B)T .
p,m n,q
This can be shown by introducing an arbitrary n×q matrix C:
T (A⊗B)T vec(C) = T (A⊗B)vec(C⊤)
p,m n,q p,m
⊤ ⊤
= Tp,mvec(BC A )
= vec(ACB⊤)
= (B⊗A)vec(C).
This implies that ((B ⊗ A) − Tp,m(A ⊗ B)Tn,q)vec(C) = 0. Because C is
arbitrary, the desired result must hold.
An immediate corollary to the above result is that
(A⊗B)T =T (B⊗A).
n,q m,p
It is also useful to note that T =T =I . Thus, if A is 1×n then
1,m m,1 m
(A⊗B)T =(B⊗A). When working with derivatives of scalars this can
n,q
result in considerable simplification.
Turning now to calculus, define the derivative of a function mapping
n m
ℜ →ℜ asthem×nmatrixofpartial derivatives:
[Df] = ∂fi(x).
ij ∂x
j
For example, the simplest derivative is
dAx =A.
dx
Using this definition, the usual rules for manipulating derivatives apply
naturally if one respects the rules of matrix conformability. The summation
rule is obvious:
D[αf(x)+βg(x)] = αDf(x)+βDg(x),
3
where α and β are scalars. The chain rule involves matrix multiplication,
n m
which requires conformability. Given two functions f : ℜ → ℜ and
p n
g : ℜ → ℜ , the derivative of the composite function is
D[f(g(x))] = f′(g(x))g′(x).
Notice that this satisfies matrix multiplication conformability, whereas the
expression g′(x)f′(g(x)) attempts to postmultipy an n × p matrix by an
⊤
m×nmatrix. To define a product rule, consider the expression f(x) g(x),
n m
where f,g : ℜ → ℜ . The derivative is the 1×n vector given by
⊤ ⊤ ′ ⊤ ′
D[f(x) g(x)] = g(x) f (x)+f(x) g (x).
Notice that no other way of multiplying g by f′ and f by g′ would ensure
conformability. A more general version of the product rule is defined below.
The product rule leads to a useful result about quadratic functions:
⊤
dx Ax ⊤ ⊤ ⊤ ⊤ ⊤
dx =x A+x A =x (A+A ).
⊤ ⊤
When A is symmetric this has the very natural form dx Ax/dx = 2x A.
These rules define derivatives for vectors. Defining derivatives of matri-
ces with respect to matrices is accomplished by vectorizing the matrices, so
dA(X)/dX is the same thing as dvec(A(X))/dvec(X). This is where the
the relationship between the vec operator and Kronecker products is useful.
⊤
Consider differentiating dx Ax with respect to A (rather than with respect
to x as above):
⊤ ⊤ ⊤
dvec(x Ax) d(x ⊗x )vec(A) ⊤ ⊤
dvec(A) = dvec(A) =(x ⊗x )
(the derivative of an m × n matrix A with respect to itself is Imn). n
A more general product rule can be defined. Suppose that f : ℜ →
m×p n p×q n m×q
ℜ and g : ℜ → ℜ , so f(x)g(x) : ℜ → ℜ . Using the relationship
between the vec and Kronecker product operators
⊤
vec(I f(x)g(x)I ) = (g(x) ⊗I )vec(f(x)) = (I ⊗f(x))vec(g(x)).
m q m q
Anatural product rule is therefore
⊤ ′ ′
Df(x)g(x) = (g(x) ⊗I )f (x)+(I ⊗f(x))g (x).
m q
This can be used to determine the derivative of dA⊤A/dA where A is
m×n.
⊤ ⊤ ⊤ ⊤ ⊤
vec(A A) = (I ⊗A )vec(A) = (A ⊗I )vec(A ) = (A ⊗I )T vec(A).
n n n m,n
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