1. Matrices
MSU CSE 260

2. Outline Introduction Matrix Arithmetic:
Sum, Product
Transposes and Powers of Matrices
Identity matrix, Transpose, Symmetric matrices
Zero-one Matrices:
Join, Meet, Boolean product
Exercise 2.6

3. Introduction Definition A matrix is a rectangular array of numbers.
element in ith row, jth column
Also written as A=aij
m rows
mn matrix
n columns
When m=n, A is called a square matrix.

4. Matrix Equality Definition Let A and B be two matrices.
A=B if they have the same number of rows and columns, and every element at each position in A equals element at corresponding position in B.

5. Matrix Addition, Subtraction
Let A = aij , B = bij be mn matrices. Then:
A + B = aij + bij, and A - B = aij - bij

6. Matrix Multiplication
Let A be a mk matrix, and B be a kn matrix,

7. Matching Dimensions
To multiply two matrices, inner numbers must match:
Otherwise,
not defined.
23 34
24 matrix
have to be equal
24
23
34

8. Multiplicative Properties
Note that just because AB is defined, BA may not be.
Example If A is 34, B is 46, then AB=36, but
BA is not defined (46 . 34).
Even if both AB and BA are defined, they may not have
the same size. Even if they do, matrices do not commute.

9. Efficiency of Multiplication
34
23
a11b12 + a12b22 + a13b32 = c12
Takes 3 multiplications, and 2 additions for each element.
This has to be done 24 (=8) times (since product matrix is 24). So 243 multiplications are needed.
(mk) (kn) matrix product requires m.k.n multiplications.

10. Best Order? Let A be a 2030 matrix, B 3040, C 4010. (AB)C or A(BC)?
(2030 3040) 4010
32000
2030 (3040 4010)
18000
So, A(BC) is best in this case.

11. Identity Matrix The identity matrix has 1’s down the diagonal, e.g.:
For a mn matrix A, Im A = A In
mm mn = mn nn

12. Inverse Matrix Let A and B be nn matrices.
If AB=BA=In then B is called the inverse of A, denoted B=A-1.
Not all square matrices are invertible.

13. Use of Inverse to Solve Equations
Please note that a-1j is NOT necessarily (aj)-1.

14. Transposes of Matrices
Flip across diagonal
Transposes are used frequently in various algorithms.

15. Symmetric Matrix A is called symmetric.
is symmetric. Note, for A to be
symmetric, is has to be square.
is trivially symmetric...

16. Examples

17. Power Matrix
For a nn square matrix A, the power matrix is defined as:
Ar = A  A  …  A
r times
A0 is defined as In.

18. Zero-one Matrices All entries are 0 or 1. Operations are  and .
Boolean product is defined using:
 for multiplication, and
 for addition.

19. Boolean Operations Terminology is from Boolean Algebra.Think
“join” is “put together”, like union, and
“meet” is “where they meet”, or intersect.

20. Boolean Product Since this is “or’d”, you can stop when you find a ‘1’
(Should be a ‘dot’)
Since this is “or’d”, you
can stop when you find
a ‘1’

21. Boolean Product Properties
In general, A  B  B  A
Example

22. Boolean Power
A Boolean power matrix can be defined in exactly the same way as a power matrix. For a nn square matrix A, the power matrix is defined as:
A[r] = A  A  …  A
r times
A[0] is defined as In.

23. Exercise 2.6