permutation matrices
permutation matrices：置换矩阵。
在数学上，特别是在矩阵理论中，置换矩阵是一个方形二进制矩阵，它在每行和每列中只有一个1，而在其他地方则为0。
设P 是一个 m×n 的 (0,1) 矩阵，如果 m≤n且 PP′=E，则称 P为一个 m×n的置换矩阵。其中P′是P的转置矩阵，E是m阶单位方阵。
定理 1 当 m≦n时，一个 m×n 的(0,1) 矩阵P为置换矩阵的充要条件是P的每一行恰有一个 1，每一列恰有一个 1。
置换矩阵在数学中的矩阵论里，置换矩阵是一种系数只由0和1组成的方块矩阵。置换矩阵的每一行和每一列都恰好有一个1，其余的系数都是0。在线性代数中，每个n阶的置换矩阵都代表了一个对n个元素（n维空间的基）的置换。当一个矩阵乘上一个置换矩阵时，所得到的是原来矩阵的横行（置换矩阵在左）或纵列（置换矩阵在右）经过置换后得到的矩阵。
Tutorial
Combination
An arrangement of r objects,
WITHOUT regard to ORDER and without repetition,
selected from n distinct objects is called a
combination of n objects taken r at a time.
The number of such combinations is denoted by
The difference between combinations and permutations is in
combinations you are counting groups (order is not important) and
in permutations you are counting different ways to arrange items
with regard to order.
The n and the r mean the same thing in both the permutation and
combinations, but the formula differs. Note that the combination
has an extra r! in its denominator.
Permutation An ORDER
of arrangements of r objects, without repetition,
selected from n distinct objects is called a permutation of
n objects taken r at a time, and is denoted as
In other words, when you need to count the number of ways
Math 233 Hessians Fall 2001
and Unconstrained Optimization
The Big Picture: Second derivatives, whether in single – or multi – variable calculus,
measure the rate of change in slopes (i.e. the curvature of the function f). What makes
problems harder in multivariable calc is that we have slopes in infinitely many directions
(directional derivatives). So, we somehow need to examine how these infinite number of
slopes change to help us determine the curvature and shape of the function f near critical
points.
This brings to mind something like second directional derivatives. We summarized the
information about slopes by creating a vector of partial derivatives – the gradient. In a
similar way, we can help to summarize known information about the rate of change of
slopes by creating a matrix of second partial derivatives – the Hessian. So here is what we
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Package‘MDR’
December13,2010
TypePackage
TitleDetectgene-geneinteractionsusingmultifactordimensionalityreduction
Version1.0
Date2010-07-16
AuthorStaceyWinham
MaintainerStaceyWinham
DescriptionPerformsmultifactordimensionalityreduction(MDR)to
detectpotentialgene-geneinteractionsincase-controlstudies.
Dependslattice
LicenseGPL-2
LazyLoadyes
RepositoryCRAN
Date/Publication2010-12-1320:18:31
Rtopicsdocumented:
MDR-package........................................2
boot.error..........................................3
compare...........................................5
mdr.............................................5
mdr.3WS..........................................7
mdr.ca.adj..........................................10
mdr.cv............................................11
mdr.hr............................................14
mdr1.............................................15
