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Matrix algebra for beginners, Part I matrices ...

Matrix algebra for beginners, Part Imatrices, determinants, inversesJeremy GunawardenaDepartment of Systems BiologyHarvard Medical School200 Longwood Avenue, Cambridge, MA 02115, January 2006 Contents1 Introduction12 Systems of linear equations13 matrices and Matrix multiplication24 matrices and complex numbers55 Can we use matrices to solve linear equations?66 Determinants and the inverse matrix77 Solving systems of linear equations98 Properties of determinants109 Gaussian elimination1111 IntroductionThis is a Part I of an introduction to the Matrix algebra needed for the Harvard Systems Biology101 graduate course. Molecular systems are inherently many dimensional there are usually manymolecular players in any biological system and linear algebra is a fundamental tool for thinkingabout many dimensional systems. It is also widely used in other areas of biology and will describe the main concepts needed for the course determinants, Matrix inverses, eigenvaluesand eigenvectors and try to explain where the concepts come from, why they are important andhow they are used.

2 Systems of linear equations Matrices first arose from trying to solve systems of linear equations. Such problems go back to the very earliest recorded instances of mathematical activity. A Babylonian tablet from around 300 BC states the following problem1: There are two fields whose total area is 1800 square yards. One produces grain at the

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SUPPORTING INFORMATION - Harvard University

vcp.med.harvard.edu

This note provides supporting information for the paper “A linear framework for time-scale separation in nonlinear biochemical systems”. It should be read in conjunction with the paper. The material is roughly organised in the order in which it appears in the paper

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Matrix algebra for beginners, Part I matrices ...

vcp.med.harvard.edu

Matrix algebra for beginners, Part I matrices, determinants, inverses Jeremy Gunawardena Department of Systems Biology Harvard Medical School 200 Longwood Avenue, Cambridge, MA 02115, USA

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Advanced Review Post-translational modification: nature’s ...

vcp.med.harvard.edu

Advanced Review Post-translational modification: nature’s escape from genetic imprisonment and the basis for dynamic information encoding Sudhakaran Prabakaran,1 Guy Lippens,2 Hanno Steen3 and Jeremy Gunawardena1∗ We discuss protein post-translational modification (PTM) from an information

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1.3 Solving Systems of Linear Equations: Gauss-Jordan ...

www.math.tamu.edu

1.3 Solving Systems of Linear Equations: Gauss-Jordan Elimination and Matrices We can represent a system of linear equations using an augmented matrix. In general, a matrix is just a rectangular arrays of numbers. Working with matrices allows us to not have to keep writing the variables over and over.

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11. LU Decomposition - University of California, Davis

www.math.ucdavis.edu

Example Linear systems associated to triangular matrices are very easy to solve by back substitution. a b 1 0 c e!)y= e c:x= a (1 be) 0 B @ 1 0 0 d a 1 0 e b c 1 f 1 C A)x= d: y= e ad; z= f bd c(e ad) For lower triangular matrices, back substitution gives a quick solution; for upper triangular matrices, forward substitution gives the solution. 1

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9. Properties of Matrices Block Matrices

www.math.ucdavis.edu

Linear Systems Redux Recall that we can view a linear system as a ma-trix equation MX= V; with Man r kmatrix of coe cients, xa k 1 matrix of unknowns, and V an r 1 matrix of constants. If Mis a square matrix, then the number of equations (r) is the same as the number of unknowns (k), so we have hope of nding a single solution.

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FACTORIZATION of MATRICES - University of Texas at Austin

web.ma.utexas.edu

method of elimination for solving systems of linear equation. A A A ... elementary matrices, is lower triangular with entries on the diagonal and is upper triangular. Fundamental Theorem 2 is the version that's most often used in large scale computations. But rather than

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www.math.iit.edu

systems of linear equations). The basic idea is to use left-multiplication of A ∈Cm×m by (elementary) lower triangular matrices, L 1,L 2, ...

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Systems of Linear Equations

people.ucsc.edu

3.5 Systems of Linear Equations in Three Variables and Applications 3.6 Solving Systems of Linear Equations by Using Matrices 3.7 Determinants and Cramer’s Rule 177 IA 3 miL2872X_ch03_177-254 09:22:2006 02:15 PM Page 177 CONFIRMING PAGES. 178 Chapter 3 Systems of Linear Equations IA 1. Solutions to Systems of Linear Equations

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www.cs.unc.edu

optimal linear filter. (8) The Dual Problem. The new formulation of the Wiener problem brings it into contact with the growing new theory of control systems based on the “state” point of view [17–24]. It turns out, surprisingly, that the Wiener problem is the dual of the noise-free optimal regulator problem, which has been solved

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Lecture 17 Perron-Frobenius Theory - Stanford University

stanford.edu

nonnegative matrices arise in many fields, e.g., • economics • population models • graph theory • Markov chains • power control in communications • Lyapunov analysis of large scale systems Perron-Frobenius Theory 17–3

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Transfer Functions - cds.caltech.edu

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linear time-invariant system by its response to sinusoidal signals. The idea goes back to Fourier, who introduced the method to investigate propagation of heat in metals. Frequency response gives an alternative way of viewing dynamics. One advantage is that it is possible to deal with systems of very high order, even inflnite.

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