1. Lecture 11 Matrices and Linear Algebra with MATLAB
Dr .Qi Ying

2. Objectives Review Matrix Operations Review Linear Algebra
Linear systems of equations
Matrix operations
Solve linear systems of equations in MATLAB

3. Matrix Notation Principle diagonal

4. Matrix Notation
Row vector and column vector

5. Matrix Terminology
Diagonal matrix: square matrix where all elements off the main diagonal are zero
Identity matrix: diagonal matrix where all elements on the main diagonal are equal to 1

6. Matrix Terminology
Upper/lower triangular matrix

7. Matrix Terminology
Banded matrix
Tridiagonal matrix: Band width = 3

8. Matrix Operation
Transpose: exchange rows for columns

9. Matrix Terminology
Symmetric matrix: aij=aji
or AT=A

10. Matrix Operation Addition/subtraction Must be of the same size
Perform operation element by element

11. Matrix Operation Multiplication with a scalar
The scalar is multiplied with each element to form a new matrix

12. Matrix Operation Multiplication with matrix
Number of columns in the first matrix must equal to the number of rows in the second matrix
row1
Column 2

13. Matrix Operation In general, matrix multiplication is NOT commutative:
For square matrix:

14. Matrix Operation Matrix inversion: for square matrix
B is defined as the inverse of matrix A, or
[B]=[A]-1

15. Matrix Operation
Matrix inversion for 2x2 matrix

16. Linear systems of equations

17. Linear systems of equations
Solution of linear systems using MATLAB
>> A=[5 3 2; 4 3 7; 7 5 1];
>> b=[7 9 5]';
>> x=inv(A)*b
x =
3.1852
1.0370

18. Linear systems of equations
Check the results:
>> x=A\b
x =
3.1852
1.0370
>> A*x
ans =
7.0000
9.0000
5.0000

19. Matrix Operation
Matrix augmentation: adding one or more columns to the original matrix.
Useful when we want to apply identical row operations on two matrices.