ANIMAL GENETICS & BREEDING Biometrical Techniques
Description: ANIMAL GENETICS BREEDING Biometrical Techniques in Animal Breeding Course No. AGB 605 UNIT - II Lecture 9 Introduction to Matrix Algebra Dr K G Mandal Department of Animal Genetics Breeding Bihar Veterinary College, Patna Bihar
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slide1. ANIMAL GENETICS & BREEDING
Biometrical Techniques in Animal Breeding
Course No. AGB – 605
UNIT - II Lecture – 9
Introduction to Matrix Algebra
Dr K G Mandal Department of Animal Genetics & Breeding Bihar Veterinary College, Patna Bihar Animal Sciences University, Patna<br>
slide2. Introduction to Matrix Algebra Science of today is becoming increasingly quantitative in nature.
Scientists are being confronted with large volumes of numerical data gathered from their laboratories, field experiments and surveys.
Mere collection and recording of data achieves nothing until or unless those data are analysed and interpreted.
Mathematics is made great use of in describing this analysis and interpretation, and one of the most important and useful branches of mathematics for this purpose is matrix algebra.<br>
slide3. It is useful not only for simplifying description and development of many analysis methods but also in organising computer techniques to execute those methods and to present the results.
As a branch of mathematics, it dates back more than a century, but its application in today’s world are widespread, particularly in statistics.
Matrices are simply rectangular arrays of numbers arranged in rows and columns, matrix algebra is the algebra of those arrays.<br>
slide4. Application:
Useful in the field of population dynamics to investigate the distribution of individuals according to their age and sex.
Calculation of transition probabilities over a range of time intervals.
continued on the next page<br>
slide5. 3. In the field of Animal Breeding.
For genetic improvement of farm animals, the animals are selected by using a selection methods known as selection index.
The numerical score of selection index is obtained through solving partial regression coefficients (b’s) for all the characters included in the index.
The partial regression coefficients are solved through the technique of matrix algebra.
SI = b1X1 + b2X2 + b3X3 + . . . . + bnXn<br>
slide9. Instead of using numbers, “ i “ is used as subscript to denote the row and “ j “ is used to denote column, where i = 1, 2, 3, and j = 1, 2, 3, 4.
Thus, the element aij means the element is located at ith row and jth column.
Accordingly, the element a11 is located at first row and first column of the matrix, A.
Similarly, the matrix, A = [ aij ] where, i = 1, 2, 3, and j = 1, 2, 3, 4 like this.<br>
slide13. 7. Subdiagonal elements:
The elements of a square matrix that lie in a line parallel to and just below the diagonal are referred as subdiagonal elements. e.g. 5, 3, 5.
A = 1 7 0 7
5 2 9 1
0 3 4 6
8 0 5 7<br>
slide19. General notation:
Matrices are usually denoted by boldface capital letters and their elements by the small letter with appropriate subscripts.
Contrary to this, vectors are denoted by boldface small letters, often from the end of the alphabets, using the prime as superscript to distinguish a row vector from a column vector.
Thus, x is a column vector and x’ is a row vector.<br>
slide21. Special Matrices
Symmetric matrices:
A square matrix is said to be symmetric when it equals to its transpose.
Exchange of rows with corresponding columns of a matrix is known as transposition and the new matrix is the transpose of original matrix. It is denoted with prime to differentiate from original matrix.
A is symmetric when A = A′ , with aij = aji for i,j = 1, 2, 3 . . . . . r for Arxr
Example:
A = 1 2 3
2 1 4 = aij for i,j = 1, 2, 3
3 4 1<br>
slide22. 2. Orthogonal matrix:
A matrix is said to be orthogonal if the product of a matrix A with its transpose A’ equals an identity matrix.
That is AA’ = I = A’A
A = 1 2 3 1 2 3 1 0 0
2 1 4 x 2 1 4 = 0 1 0
3 4 1 3 4 1 0 0 1<br>
slide23. THANK YOU<br>
Biometrical Techniques in Animal Breeding
Course No. AGB – 605
UNIT - II Lecture – 9
Introduction to Matrix Algebra
Dr K G Mandal Department of Animal Genetics & Breeding Bihar Veterinary College, Patna Bihar Animal Sciences University, Patna<br>
slide2. Introduction to Matrix Algebra Science of today is becoming increasingly quantitative in nature.
Scientists are being confronted with large volumes of numerical data gathered from their laboratories, field experiments and surveys.
Mere collection and recording of data achieves nothing until or unless those data are analysed and interpreted.
Mathematics is made great use of in describing this analysis and interpretation, and one of the most important and useful branches of mathematics for this purpose is matrix algebra.<br>
slide3. It is useful not only for simplifying description and development of many analysis methods but also in organising computer techniques to execute those methods and to present the results.
As a branch of mathematics, it dates back more than a century, but its application in today’s world are widespread, particularly in statistics.
Matrices are simply rectangular arrays of numbers arranged in rows and columns, matrix algebra is the algebra of those arrays.<br>
slide4. Application:
Useful in the field of population dynamics to investigate the distribution of individuals according to their age and sex.
Calculation of transition probabilities over a range of time intervals.
continued on the next page<br>
slide5. 3. In the field of Animal Breeding.
For genetic improvement of farm animals, the animals are selected by using a selection methods known as selection index.
The numerical score of selection index is obtained through solving partial regression coefficients (b’s) for all the characters included in the index.
The partial regression coefficients are solved through the technique of matrix algebra.
SI = b1X1 + b2X2 + b3X3 + . . . . + bnXn<br>
slide9. Instead of using numbers, “ i “ is used as subscript to denote the row and “ j “ is used to denote column, where i = 1, 2, 3, and j = 1, 2, 3, 4.
Thus, the element aij means the element is located at ith row and jth column.
Accordingly, the element a11 is located at first row and first column of the matrix, A.
Similarly, the matrix, A = [ aij ] where, i = 1, 2, 3, and j = 1, 2, 3, 4 like this.<br>
slide13. 7. Subdiagonal elements:
The elements of a square matrix that lie in a line parallel to and just below the diagonal are referred as subdiagonal elements. e.g. 5, 3, 5.
A = 1 7 0 7
5 2 9 1
0 3 4 6
8 0 5 7<br>
slide19. General notation:
Matrices are usually denoted by boldface capital letters and their elements by the small letter with appropriate subscripts.
Contrary to this, vectors are denoted by boldface small letters, often from the end of the alphabets, using the prime as superscript to distinguish a row vector from a column vector.
Thus, x is a column vector and x’ is a row vector.<br>
slide21. Special Matrices
Symmetric matrices:
A square matrix is said to be symmetric when it equals to its transpose.
Exchange of rows with corresponding columns of a matrix is known as transposition and the new matrix is the transpose of original matrix. It is denoted with prime to differentiate from original matrix.
A is symmetric when A = A′ , with aij = aji for i,j = 1, 2, 3 . . . . . r for Arxr
Example:
A = 1 2 3
2 1 4 = aij for i,j = 1, 2, 3
3 4 1<br>
slide22. 2. Orthogonal matrix:
A matrix is said to be orthogonal if the product of a matrix A with its transpose A’ equals an identity matrix.
That is AA’ = I = A’A
A = 1 2 3 1 2 3 1 0 0
2 1 4 x 2 1 4 = 0 1 0
3 4 1 3 4 1 0 0 1<br>
slide23. THANK YOU<br>