Lecture 9: Submatrices and Some Special Matrices

Transcription

1 Lecture 9: Submatrices and Some Special Matrices Winfried Just Department of Mathematics, Ohio University February 2, 2018

2 Submatrices A submatrix B of a given matrix A is any matrix that can be obtained by removing some rows and/or columns from A The rows and columns that remain do not need to be adjacent in A Consider A = Each of the following is a submatrix of A: B 1 = B 2 = B 3 = [ ]

3 Homework Homework 24: Consider the matrix A = Which of the following are submatrices of A? B 1 = [ 2 3 ] B 4 = [ 13 ] B 5 = B 2 = [ 6 7 ] [ ] B 3 = [ 1 3 ]

4 Doing nothing with numbers and matrices Adding zero to a number a does nothing: 0 + a = a + 0 = a Similarly, multiplying a number a by 1 does nothing: 1 a = a 1 = a We have already seen the analogue of 0 for matrices: Let O m n = = [0] m n Then O m n + A = A + O m n = A for every matrix A of order m n We call O m n the zero matrix of order m n When the order is implied by the context, we simply write O instead of O m n

5 The matrix version of 1 Is there a matrix 1 such that 1A = A and A1 = A whenever the products make sense? Yes! Let 1 = = [1] n n = I We call I the identity matrix of order n n When we want to specify the order, we write I n The notation I n n would be more consistent with the one we use for O, but we can simplify it here as I must always be a square matrix

6 The matrix I is the left multiplicative unity I n A = a 11 a 1j a 1p a i1 a ij a ip a n1 a nj a np = a 11 a 1j a 1p a i1 a ij a ip a n1 a nj a np = A Thus I n A = A for every matrix A or order n p

7 The matrix I is the right multiplicative unity AI n = a 11 a 1j a 1n a i1 a ij a in a m1 a nj a mn = a 11 a 1j a 1n a i1 a ij a in a m1 a mj a mn = A Thus AI n = A for every matrix A or order m n

8 The do-nothingness of I In terms of matrix multiplication, doing nothing means don t change anything We have already seen I 2 in our weathercom example: [ ] 1 0 The transition probability matrix P = I 2 = implies that the 0 1 state of the weather never changes Similarly, the transformation T I : R 2 R 2 is the identity map [ ] [ ] [ ] 1 0 x x T I ( v) = = = v 0 1 y y that leaves every vector unchanged You can implement it by doing nothing to your sheet of paper

9 Positive integer powers of a square matrix Let A be a square matrix For a positive integer r, the r-th power of A is defined as A r = AAA AA, where the product on the right contains r terms By the Associativity Law, when r = p + q and p, q > 0, we can group the product into p factors followed by q factors as follows: A r = A p+q = (AA A)(AA A) = A p A q, which is exactly the same rule as a p+q = a p a q for powers of numbers In particular, A p+1 = A p A 1 = A p A Similarly, the law (A r ) p = A rp holds for square matrices in analogy with the law (a r ) p = a rp for numbers

10 Powers A r when r is an integer 0 Can we meaningfully define A r for integers r 0 so that the laws A p+q = A p A q and (A r ) p = A rp still hold? In analogy with a 0 = 1 we let: A 0 = I n when A has order n n Then A q = IA q = A 0 A q = A 0+q and (A 0 ) p = I p = I = A 0 = A 0p, so this works just fine Negative powers of matrices are more problematic A 1 would need to satisfy A 1 A = A 1 A 1 = A 0 = I We will see later in this course that not every square matrix A has such a multiplicative inverse A 1 The situation is not entirely unlike the one for numbers; the only difference is that while a = 0 is the only number without multiplicative inverse a 1 = 1 a, for any given n > 1 there are infinitely many n n matrices A for which A 1 does not exist

11 Is multiplication by a scalar a special case of a matrix product? Yes, but you need to interpret the scalar as a matrix of the right order first [λ] 1 1 doesn t usually work, unless you want to compute λ w for a 1 n row vector w, or vλ for an m 1 column vector v In general, if A is of order m n, then λa = (λi m )A, Aλ = A(λI n ), where λ λ 0 λi = = Iλ 0 0 λ

12 Let s see how this works (λi m )A = λ λ λ a 11 a 1j a 1n a i1 a ij a in a m1 a mj a mn = λa 11 λa 1j λa 1n λa i1 λa ij λa in λa m1 λa mj λa mn = λa

13 Diagonal matrices A square matrix A is diagonal if a ij = 0 whenever i j, that is, when A is of the form: λ λ A = 0 0 λ λ n Every matrix λi is diagonal, but there are other examples, like: A = B = [ ] C =

14 Products of diagonal matrices Homework 25: Show that the product of two diagonal matrices of the same order is again a diagonal matrix More specifically, derive the following general formula: λ κ λ κ 2 0 = 0 0 λ n 0 0 κ n λ 1 κ λ 2 κ λ n κ n

15 Triangular matrices A square matrix U = [u ij ] n n is upper-triangular if u ij = 0 whenever i > j, that is, when u ij sits below the main diagonal A square matrix L = [l ij ] n n is lower-triangular if l ij = 0 whenever i < j, that is, when l ij sits above the main diagonal Consider the following examples: U = L = The product of two upper-triangular matrices of the same order is again an upper-triangular matrix The product of two lower-triangular matrices of the same order is again a lower-triangular matrix

16 To see how this works: Assume i > j and multiply two upper-triangular matrices: u 11 u 1j u 1n v 11 v 1j v 1n 0 u ij = 0 u in 0 v ij = 0 v in 0 0 u nn 0 0 v nn c 11 c 1j c 1n = 0 c ij = 0 c in 0 0 c nn Each product of blue or magenta terms has a factor u ik = 0, Each product of red or magenta terms has a factor v kj = 0

17 LU-decompositions It is often useful to represent matrices as products of special matrices that are easier to work with Such products are called matrix decompositions The following theorem that we will prove later in this course gives an example Theorem Let A be any n n square matrix Then their exists exactly one pair (L, U) of matrices, called the LU-decomposition of A, such that: Both L and U of the same order n n as A L is lower-triangular U is upper-triangular u ii = 1 for every element u ii of the main diagonal of U A = LU

18 Some practice problems Homework 26: For each of the following statements, determine whether it is true or false If the statement is true, prove it If the statement is false, give a counterexample (a) If A is an n n diagonal matrix, then a ii 0 for all i = 1,, n (b) If U is an upper-triangular matrix of order n n, then U T is a lower-triangular matrix of order n n (c) If U is an upper-triangular matrix of order n n and L is a lower-triangular matrix of order n n, then U + L is a symmetric matrix (d) If U is an upper-triangular matrix of order 3 3, then U 3 is the zero matrix O of order 3 3 (e) If U is an upper-triangular matrix of order 3 3 that has only zeros on the main diagonal, then U 3 is the zero matrix O of order 3 3