1. 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 )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 matrix with two rows and four columns 1234 5678 A matrix with two rows and three columns 123 567 A matrix with two rows and two columns 12 56

3. A matrix with two rows and four columns 1234 5678 A matrix with two rows and two columns 15 26 37 48

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

5. The size of this matrix is 2 rows by 3 columns (we say ‘2 by 3’ and write ‘2 x 3’) x 1,1 x 1,2 x 1,3 x 1,2 x 2,2 x 2,3 We use two indices to identify an entry in the matrix: a row and column index row 1 column 3 Entry in row 1, column 3: x 1,3 [X] 1,3 (X) 1,3 Matrix symbols: Capital letters ‘X’ or underlined Capital letters ‘X’ 1234 5678 X entry (X) 2,3 = 7

6. The size of this matrix is 2 rows by 3 columns (we say ‘2 by 3’ and write ‘2 x 3’) x 1,1 x 1,2 x 1,3 x 1,2 x 2,2 x 2,3 We use two indices to identify an entry in the matrix: a row and column index ‘(‘ 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 square matrix of size n by n with n=3 A rectangular matrix of size m by n with m=3 and n=6 (m<n) A rectangular matrix of size m by n with m=4 and n=3 (m>n)

8. 1234 5678 Multiplication with a scalar: 2 2468 10121416 = c X = Z Size: m x n m x n (Z) i,j = c(x) i,j (For all i and j)

9. 1234 5678 0100 001 + 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) Addition of Matrices: 1334 5669 =

10. 1234 5678 X T = Y Size: m x n n x m (X) i,j = (Y) j,i (For all i and j) Transpose of a Matrix 15 26 37 48 = T

11. Matrix Multiplication: A B = 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 as the right matrix B.

12. Matrix Multiplication: A B = C 101 011 15 26 37 48 Size: 4 x 2 2 x 3 4 x 3 = ?

13. Matrix Multiplication: A B = C 101 011 15 26 37 48 Size: 4 x 2 2 x 3 4 x 3 = 1*1+0*5 Column 1 Row 1 Vector dot product

14. Matrix Multiplication: A B = C 101 011 15 26 37 48 Size: 4 x 2 2 x 3 4 x 3 = 1*1+0*5 3*0+7*1 Column 2 Row 3

15. Matrix Multiplication: A B = C 101 011 15 26 37 48 Size: 4 x 2 2 x 3 4 x 3 = 1*1+0*51*1+5*1 3*0+7*1 Column 3 Row 1

16. Matrix Multiplication: A B = C 101 011 15 26 37 48 Size: 4 x 2 2 x 3 4 x 3 = 156 268 3710 4812

17. Matrix Multiplication: A B = C Size: m x n n x k m x k Note: A B is not equal B A ! 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.

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