Lecture 18 &amp; 19 Matrix Arithmetic for Use in the Biological Sciences

# Lecture 18 & 19 Matrix Arithmetic for Use in the Biological Sciences

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I OWA S TATE U NIVERSITY Department of Animal Science Matrix Arithmetic for Use in the Biological Sciences Animal Science 500 Lecture No. 18 & 19 November 9, 2010

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I OWA S TATE U NIVERSITY Department of Animal Science Matrix Arithmetic and Algebra References 1. Matrix Algebra for the Biological Sciences – S. R. Searle. 1966. John Wiley & Sons, New York, N.Y. (This book is older and out of print. For those interested in a copy, there are used copies available from resellers from various internet sites.) 2. Linear Algebra for Dummies – M. J. Sterling. 2009. John Wiley & Sons, New York, N.Y. (This book is relatively new and can be purchased new or used from numerous resellers found on the internet.)
I OWA S TATE U NIVERSITY Department of Animal Science Matrix Arithmetic and Algebra – Why? “The simulation of physiological systems requires a mathematical base and in most cases a large computer.” (W. J. Dixon, 1966) Attempting to turn what we observe or what is occurring physiologically or what is commonly referred to as a “phenotype” into a mathematical model. Explain experimental results Explain various environmental factor Today biological sciences are very much quantitative whereas years ago it was more descriptive

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I OWA S TATE U NIVERSITY Department of Animal Science What is Matrix Algebra Is in a way a shorthand notation for the language of mathematics Provides the ability to deal with many numbers and / or equations simultaneously A matrix is simply a rectangular array of numbers set out in rows and columns. Is frequently used in organizing the presentation of numerical data that will be handled in some way mathematically. Common examples Animal breeding – solving equations to estimate variance components and breeding values Solving simultaneous equations – nutritional nutrient balancing Data analysis – any procedure that involves linear equations involves the use of matricies.
I OWA S TATE U NIVERSITY Department of Animal Science Matrix and Regression Analysis An example that illustrates the wide spread use of matrix and matrix algebra y = b0 + b1 x1 + b2 x2 + … + bk xk Where there are numerous observations on the variable y and on each of the k variables x1, x2, …. xk. b’s values can be obtained when X’Xb = X’y as b = (X’X)-1 X’y is solved. Where X and y are both matrices representing all of the observations in the x and y variables respectively and b represent the series of b’s

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I OWA S TATE U NIVERSITY Department of Animal Science Matrix Algebra Is a mathematical procedure for many problems of any size (small and large) can be described. Hence, size does not affect the understanding of the procedures just the amount of calculating or computer time required to solve the equations.
I OWA S TATE U NIVERSITY Department of Animal Science General description of a Matrix A matrix is an aid in organizing data.

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Lecture 18 & 19 Matrix Arithmetic for Use in the Biological Sciences

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