Optimization Models
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1. Introduction
1.1. Overview
1.2. Optimization Models
2. Linear Algebra
2.1. Vectors
2.2. Matrices
2.3. Linear Equations
2.4.  Least-Squares
2.5. Eigenvalues
2.6  Singular Values
3. Convex Models
3.1. Convexity
3.2. LP and QP
3.3. SOCP
3.4. Robust LP
3.5. GP
3.6. SDP
3.7. Non-Convex Models
4. Duality
4.1. Weak Duality
4.2. Strong Duality
4.3.  Applications
5. Case Studies
5.1. Senate Voting
5.2. Antenna Arrays
5.3. Localization
5.4. Circuit Design

Permutation matrices

A n times n matrix P is a permutation matrix if it is obtained by permuting the rows or columns of an n times n identity matrix according to some permutation of the numbers 1 to n. Permutation matrices are orthogonal (hence, their inverse is their transpose: P^{-1} = P^T) and satisfy P^2=P.

For example, the matrix

P = left( begin{array}{llllll} 1&0&0&0&0&0 0&0&1&0&0&0 0&1&0&0&0&0 0&0&0&0&1&0 0&0&0&1&0&0 0&0&0&0&0&1 end{array} right)

is obtained by exchanging the columns 2 and 3, and 4 and 5, of the 6 times 6 identity matrix.

A permutation matrix allows to exchange rows or columns of another via the matrix-matrix product. For example, if we take any 5 times 6 matrix A, then AP (with P defined above) is the matrix A with columns 2,3 and 4,5 exchanged.