A pratical introduction to MAtriCes

1. A pratical introduction to MAtriCes The goal is to give an introduction to the mathematical operations with matrices. A matrix is a 2-dimensional arrangement of (real valued) data. The data entries are organized in rows and columns, just like in a spreadsheet or a table with data. More information can be found on Wikipedia. ( http://en.wikipedia.org/wiki/Matrix_%28mathematics%29 ) This brief introduction is by far not complete, It is NOT a formal mathematical introduction to Linear Algebra !

2. A pratical introduction to MAtriCes A matrix with two rows and four columns A matrix with two rows and three columns A matrix with two rows and two columns

3. A pratical introduction to MAtriCes A matrix with two rows and four columns A matrix with two rows and two columns

4. A pratical introduction to MAtriCes column 1 column 2 column 3 The size of this matrix is 2 rows by 3 columns (we say ‘2 by 3’ and write ‘2 x 3’) row 1 row 2

5. A pratical introduction to MAtriCes We use two indices to identify an entry in the matrix: a row and column index column 3 The size of this matrix is 2 rows by 3 columns (we say ‘2 by 3’ and write ‘2 x 3’) Entry in row 1, column 3: x1,3 [X] 1,3 (X) 1,3 row 1 Matrix symbols: Capital letters ‘X’ or underlined Capital letters ‘X’ X entry (X) 2,3= 7

6. A pratical introduction to MAtriCes We use two indices to identify an entry in the matrix: a row and column index The size of this matrix is 2 rows by 3 columns (we say ‘2 by 3’ and write ‘2 x 3’) ‘(‘ and ‘)’ are used to embrace the entries, when writing matrix arrays or ‘[‘ and ‘]’ or ‘|’ ‘|’ ( ) Matrix symbols: Capital letters ‘X’ or underlined Capital letters ‘X’

7. A pratical introduction to MAtriCes A rectangular matrix of size m by n with m=3 and n=6 (m<n) A square matrix of size n by n with n=3 A rectangular matrix of size m by n with m=4 and n=3 (m>n)

8. Basic Matrix Operations Multiplication with a scalar: = (Z) i,j= c(x)i,j c X = Z (For all i and j) Size: m x n m x n

9. Basic Matrix Operations Addition of Matrices: + = X + Y = Z Size: m x n m x n m x n (Z) i,j= (X)i,j+(Y)i,j (For all i and j)

10. Basic Matrix Operations Transpose of a Matrix T = XT = Y Size: m x n n x m (X) i,j= (Y)j,i (For all i and j)

11. Basic Matrix Operations Matrix Multiplication: AB = C Size: m x n n x k m x k NOTE: Matrix multiplication is only defined for two matrices when the left matrix A has the same number of columns as the right matrix B has rows! The resulting matrix has the same number of rows as the left matrix A and the same number of columns asthe right matrix B.

12. Basic Matrix Operations Matrix Multiplication: AB = C = Size: 4 x 22 x 34 x 3

13. Basic Matrix Operations Matrix Multiplication: AB = C Column 1 Vector dot product Row 1 = Size: 4 x 22 x 34 x 3

14. Basic Matrix Operations Matrix Multiplication: AB = C Column 2 = Row 3 Size: 4 x 22 x 34 x 3

15. Basic Matrix Operations Matrix Multiplication: AB = C Column 3 Row 1 = Size: 4 x 22 x 34 x 3

16. Basic Matrix Operations Matrix Multiplication: AB = C = Size: 4 x 22 x 34 x 3

17. Basic Matrix Operations Matrix Multiplication: AB = C Note: AB is not equal BA! Rule to remember: We pick from the left matrix a row vector (row i) and from the right matrix a column vector (column j), calculate the dot product between the two vectors and enter the result in the new matrix in row i, column j. Size: m x nn x km x k ci,j=

18. Errors can easily sneak into the slides. If you find a mistake, please contact me oelisontimm@albany.edu Final NOTE: Thanks! See also a 5 minute introduction: http://ed.ted.com/lessons/how-to-organize-add-and-multiply-matrices-bill-shillito