1. Matrices Matrices For grade 1, undergraduate students For grade 1, undergraduate students Made by Department of Math.,Anqing Teachers college

2. Definition. A rectangular array of numbers composed of m rows and n columns is called an matrix (read m by n matrix). We also say that the matrix A is of, or has, size. 1 Some notations

3. The elements form the i -th row of A, and the elements form the j- th column of A. We will often write for A.

4. Definition. If are matrices, then iff for i =1,2…, m and j =1,…,n.

5. Definition. If are two matrices, their sum A+B, is the matrix, where i =1,2…, m, j =1,2…,n..Matrix opertions

6. Definition. If is an matrix and r is a number then r A, the scalar multiple of A by r, is the matrix where i =1,2…, m and j =1,…,n.

7. Proposition 1. The matrices of size form a vector space under the operations of matrix addition and scalar multiplication. We denote this vector space by M mn. The dimension of the vector space M mn is not hard to compute. We take our lead from the method we used to show that dim R n =n. Introduce the matrix by the requirement.Some properties

8. Proposition 2. The vectors form a basis for M mn. Therefore dim EXAMPLE 1.

9. EXAMPLE 2.

10. 2 Matrix products Definition. If is an matrix and is an matrix, their matrix product is the matrix, where

11. Remark. Note that for the product of A and B to be defined the number of columns of A must be equal to the number of rows of B. Thus the order in which the product of A and B is taken is very important, for AB can be defined without AB being defined.

12. EXAMPLE 4. Compute the matrix product Solution. Note the answer is a matrix

13. Remark. Note that the product is not defined. EXAMPLE 5. Compute the matrix product

14. Answer. Definition. A matrix A is said to be a square matrix of size n iff it has n rows and n columns (that is the number of rows equals the number of columns equals n ).

15. Remark. It is easy to see that if A and B are square matrices of size n then the products AB and BA are both defined. However they may not be equal.. EXAMPLE 7. Let Compute the matrix products AB and BA. Solution. We have

16. and so we see that AB BA. Remark. As the preceding example shows even if AB and BA are defined we should not expect that AB=BA.

17. Notation. If A is a square matrix then AA is defined and is denoted by A 2. Similarly, is defined and denoted by.

18. EXAMPLE 8. Let Calculate. Solution. We have

19. .The rules of matrix operations (1) A+B=B+A (2) A+(B+C)=A+(B+C) (3) r(A+B)=rA+rB (4) A+0=A (5) 0A=0 (6) A+(-1)A=0 (7) (r+s)A=rA+sA (8) (A+B)·C=A·C+C·B (9) 0·A=0=A·0 (10) A·(B·C)=(A·B) ·C

20. 3 Special types of matrices Diagonal matrices.

21. Triangular matrices. A square matrix A is said to be lower triangular iff A= whereif For example is a lower triangular matrix.

22. The Zero matrix. The zero matrix is the matrix 0 all of those entries are 0. Idempotent matrices. A square matrix A is said to be idempotent iff Nilpotent matrices. A square matrix A is said to be nilpotent iff there is an integer q such.

23. Nonsingular matrices. A square matrix A is said to be invertible or nonsingular iff there exists a matrix B such that AB=I and BA=I. Denoted by. For example if then

24. A nilpotent matrix is not invertible. For suppose that A is a nilpotent matrix that is invertible. Let B be an inverse for A. Since A is nilpotent there is an integer q such that Then so If we repeat this trick q-1 times we will get

25. But then which is impossible. Symmetric and skew-symmetric matrices. A square matrix A= is said to symmetric iff for it is said to be skew-symmetric iff for

26. For example are symmetric matrices, and are skew-symmetric matrices.

27. Proposition 3 A matrix is nonsingular iff If then

28. PROOF. Suppose that Let Then

29. and therefore A is nonsingular with Suppose conversely that A is nonsingular, but that. We will deduce a contradiction. Let

30. Then computing as above This gives the equation Therefore

31. So that But then A=0 also, so So and hence 1=0, which is impossible.

32. 4 SOME EXERCISES 1. Perform the following matrix multiplications

33. 2. Which of the following matrices are nonsingular, idempotent, nilpotent, symmetric, or skew-symmetric?

34. 3. If A is an idempotent square matrix show I-2A is invertible (Hint: Idempotent correspond to projections. Interpret I-2A as a reflection. Try the case first. Then try to generalize.) Thanks!!!