System Reliability Theory: Models and Statistical Method>
Marvin Rausand,Arnljot Hoylanc
Cowriaht 02004 bv John Wilev & Sons. Inc
Appendix C
Kronecker Products
Let A be a matrix with dimension mA x nA, and B be a matrix with dimension
m g x ng. The Kroneckerproduct' of A and B,written A @B is defined as
The Kronecker product is also known as the Zehfuss product and the tensor product.
A Kronecker sum is an ordinary sum of Kronecker products. The Kronecker sum,
written A @B,is defined for square matrices A and B as
where n~ is the size of the square matrix A, ne is the size of the square matrix B, and
II is the identity matrix.
Some Properties of the Kronecker Product
'Named after the GermadPolish mathematician Leopold Kronecker (1823-1891).
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KRONECKER PRODUCTS
1. Associativity:
A€3(B€3c)=(A€3B)€3c
2. Distributivity over ordinary matrix addition:
(A
+B)€3 (e+ D)
=
A €3
c+
B €3
c+
A €3BD€3
+D
3. Compatibility with ordinary matrix multiplication:
AB@cnD= (A@C)(B€3D)
4. Compatibility with ordinary matrix inversion:
(A€3 B)-' =A-' @B-'
Further details about the Kronecker product may be found in Graham (1981). Kronecker productsare available in many mathematicaltools, like MATLABand Maple.
System Reliability Theory: Models and Statistical Method>
Marvin Rausand,Arnljot Hoylanc
Cowriaht
02004 bv John Wilev & Sons. Inc
Appendix D
Distribution Theorems
First we refer (without proof,)two important theorems from distribution theory:
Theorem D.1 Let X be a continuousrandom variablewith probability density f x (x)
and sample space S X . Furthermore let a ( x ) be strictly monotonousin x and differentiable with respect to x for all x. Then y = a(x) is a one-to-one transformation on x
with inverse x = b ( y ) which maps SX into S y ,and the density of Y = a ( X )is given
by
fY(Y)
(D.1)
= f X ( N Y ) ) Ib'(Y)l
0
The extension of this theorem to multivariate distributionsisgiven in the next theorem.
Theorem D.2 Let X I ,X2,. ..,X, be continuouslydistributedwithjoint probability
density f x ,,x2.....x,, (XI, x2, ...,x,) and sample space S X ,,x2 ,...,X , . If
yi
= ai(xl,x2,...,x,)
for i = 1,2,...,n
(D.2)
is a one-to-one transformation on the x's with inverse
Xj
= bj(y1,~ 2 , ..., Y n ) for i = 1.2,.
that maps the sample space Sx1,x2,...,X , into Sy, ,y2,...,y,,
Yi = ai(X1,X2,... ,X,)
for
..,n
(D.3)
then thejoint density of
i = 1,2,...,n
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