Introduction to MATRIX ALGEBRA © KAW © Copyrighted to Autar K. Kaw – 2002 Introduction to Matrix Algebra – Copyright – Autar K Kaw - 1- Introduction to MATRIX ALGEBRA Autar K. Kaw University of South Florida Autar K. Kaw Professor & Jerome Krivanek Distinguished Teacher Mechanical Engineering Department University of South Florida, ENB 118 4202 E. Fowler Avenue Tampa, FL 33620-5350. Office: (813) 974-5626 Fax: (813) 974-3539 E-mail: [email protected] URL: http://www.eng.usf.edu/~kaw Introduction to Matrix Algebra – Copyright – Autar K Kaw - 2- Table of Contents Chapter 1: Introduction …………………………………………..… 4 Chapter 2: Vectors ………………………………………………… 16 Chapter 3: Binary Matrix Operations …………………………… 34 Chapter 4: Unary Matrix Operations …………………………… 47 Chapter 5: System of Equations …………………………………. 64 Chapter 6: Gaussian Elimination ………………………………… 95 Chapter 7: LU Decomposition ………………………………….… 116 Chapter 8: Gauss- Siedal Method………………………………… 131 Chapter 9: Adequacy of Solutions………………………………… 145 Chapter 10: Eigenvalues and Eigenvectors …………………….. 161 Introduction to Matrix Algebra – Copyright – Autar K Kaw - 3- Chapter 1 Introduction _________________________________ After reading this chapter, you should be able to § Know what a matrix is § Identify special types of matrices § When two matrices are equal _________________________________ What is a matrix? Matrices are everywhere. If you have used a spreadsheet such as Excel or Lotus or written a table, you have used a matrix. Matrices make presentation of numbers clearer and make calculations easier to program. Look at the matrix below about the sale of tires in a Blowoutr’us store – given by quarter and make of tires. Quarter 1 Quarter 2 Quarter 3 Quarter 4 Tirestone Ø 25 20 3 2 ø Œ œ Michigan 5 10 15 25 Œ œ Copper Œº 6 16 7 27œß If one wants to know how many Copper tires were sold in Quarter 4, we go along the row ‘Copper’ and column ‘Quarter 4’ and find that it is 27. So what is a matrix? A matrix is a rectangular array of elements. The elements can be symbolic expressions or numbers. Matrix [A] is denoted by Introduction to Matrix Algebra – Copyright – Autar K Kaw - 4- Ø a a ....... a ø Œ 11 12 1n œ [ ] Œ a a ....... a œ A = 21 22 2n Œ œ M M Œ œ º a a ....... a ß m1 m2 mn [ ] Row i of [A] has n elements and is a a ....a and i1 i2 in Ø a ø Œ 1jœ a Œ œ Column j of [A] has m elements and is 2j Œ œ M Œ œ Œº a œß mj Each matrix has rows and columns and this defines the size of the matrix. If a matrix [A] has m rows and n columns, the size of the matrix is denoted by m x n. The matrix [A] [ ] may also be denoted by [A] to show that A is a matrix with m rows and n columns. mxn Each entry in the matrix is called the entry or element of the matrix and is denoted by a ij where i is the row number and j is the column number of the element. The matrix for the tire sales example could be denoted by the matrix [A] as Ø 25 20 3 2 ø [ ] Œ œ A = 5 10 15 25 Œ œ Œ œ º 6 16 7 27ß [ ] There are 3 rows and 4 columns, so the size of the matrix is 3 x 4. In the above A matrix, a = 27. 34 What are the special types of matrices? Vector: A vector is a matrix that has only one row or one column. There are two types of vectors – row vectors and column vectors. Row vector: If a matrix has one row, it is called a row vector [B]=[b b b ] 1 2KK m and ‘m’ is the dimension of the row vector. _________________________________ Introduction to Matrix Algebra – Copyright – Autar K Kaw - 5- Example Give an example of a row vector. Solution [B] = [25 20 3 2 0] is an example of a row vector of dimension 5. _________________________________ Column vector: If a matrix has one column, it is called a column vector Ø c ø Œ 1œ [C]= Œ M œ Œ œ M Œ œ º c ß n and n is the dimension of the vector. _________________________________ Example Give an example of a column vector. Solution Ø 25ø [ ] Œ œ C = 5 is an example of a column vector Œ œ Œ œ º 6 ß of dimension 3. _________________________________ Submatrix: If some row(s) or/and column(s) of a matrix [A] are deleted, the remaining matrix is called a submatrix of [A]. Example Find some of the submatrices of the matrix Introduction to Matrix Algebra – Copyright – Autar K Kaw - 6- [ ] Ø 4 6 2ø A = Œ œ º 3 - 1 2ß Solution Ø 4 6 2ø Ø 4 6 ø [ ][ ] Ø 2ø Œ œ ,Œ œ , 4 6 2, 4,Œ œ are all submatrices of [A]. Can you find other º 3 - 1 2ß º 3 - 1ß º 2ß submatrices of [A]? _________________________________ Square matrix: If the number of rows (m) of a matrix is equal to the number of columns (n) of the matrix, (m = n), it is called a square matrix. The entries a , a , . . . a are 11 22 nn called the diagonal elements of a square matrix. Sometimes the diagonal of the matrix is also called the principal or main of the matrix. _________________________________ Example Give an example of a square matrix. Solution Ø 25 20 3ø [ ] Œ œ A = 5 10 15 Œ œ Œ œ º 6 15 7ß is a square matrix as it has same number of rows and columns, that is, three. The diagonal elements of [A] are a = 25, a = 10, a = 7. 11 22 33 _________________________________ Upper triangular matrix: A mxn matrix for which a = 0, i>j is called an upper ij triangular matrix. That is, all the elements below the diagonal entries are zero. _________________________________ Introduction to Matrix Algebra – Copyright – Autar K Kaw - 7- Example Give an example of an upper triangular matrix. Solution Ø 10 - 7 0 ø [ ] Œ œ A = 0 - 0.001 6 Œ œ Œ œ º 0 0 15005ß is an upper triangular matrix. _________________________________ Lower triangular matrix: A mxn matrix for which a = 0, j > i is called a lower ij triangular matrix. That is, all the elements above the diagonal entries are zero. _________________________________ Example Give an example of a lower triangular matrix. Solution Ø 1 0 0ø [ ] Œ œ A = 0.3 1 0 Œ œ Œ œ º 0.6 2.5 1ß is a lower triangular matrix. _________________________________ Diagonal matrix: A square matrix with all non-diagonal elements equal to zero is called a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero, (a = 0, i „ j). ij _________________________________ Example Introduction to Matrix Algebra – Copyright – Autar K Kaw - 8- Give examples of a diagonal matrix. Solution Ø 3 0 0ø [ ] Œ œ A = 0 2.1 0 Œ œ Œ œ º 0 0 5ß is a diagonal matrix. Any or all the diagonal entries of a diagonal matrix can be zero. For example Ø 3 0 0ø [ ] Œ œ A = 0 2.1 0 Œ œ Œ œ º 0 0 0ß is also a diagonal matrix. _________________________________ Identity matrix: A diagonal matrix with all diagonal elements equal to one is called an identity matrix, (a = 0, i „ j; and a = 1 for all i). ij ii . _________________________________ Example Give an example of an identity matrix. Solution Ø 1 0 0 0ø Œ œ 0 1 0 0 Œ œ [A] = Œ œ 0 0 1 0 Œ œ º 0 0 0 1ß is an identity matrix. Introduction to Matrix Algebra – Copyright – Autar K Kaw - 9- _________________________________ Zero matrix: A matrix whose all entries are zero is called a zero matrix, (a = 0 for all i ij and j). _______________________________ Example Give examples of a zero matrix. Solution Ø 0 0 0ø [ ] Œ œ A = 0 0 0 Œ œ Œ œ º 0 0 0ß [ ] Ø 0 0 0 ø B = Œ œ º 0 0 0 ß Ø 0 0 0 0ø [ ] Œ œ C = 0 0 0 0 Œ œ Œ œ º 0 0 0 0ß [ ] [ ] D = 0 0 0 are all examples of a zero matrix. _________________________________ Tridiagonal matrices: A tridiagonal matrix is a square matrix in which all elements not on the major diagonal, the diagonal above the major diagonal and the diagonal below the major diagonal are zero. _________________________________ Example Give an example of a tridiagonal matrix. Solution Introduction to Matrix Algebra – Copyright – Autar K Kaw - 10-