The product of matrices

Transcription

1 Geometria Lingotto LeLing9: Matrix multiplication C ontents: The product of two matrices Proof of the Rank theorem The algebra of square matrices: the product is not commutative Rotations and elementary row operations using the product: elementary matrices The transposed matrix R ecommended exercises: Geoling 12 The product of matrices Another way of saying that the column B is a linear combination of columns A 1, A 2,, A n we use the notion of multiplication between a matrix A and a column vector C More precisely, if B is linear combination of A 1, A 2,, A n there are certain c 1, c 2,, c n such that c 1 A 1 + c 2 A n + + c n A n = B We may then think of A 1, A 2,, A n as the columns of a matrix A and assume the coefficients c 1, c 2,, c n are the entries of c 1 c 2 a column vector C = In this way the product AC of the matrix A by the vector C is by definition the column vector B c n Definition 1 Let A = (a ij be an m n matrix, and C = (c i a column vector with n components Then the product A C is the linear combination c 1 A 1 + c 2 A n + + c n A n The product A C of A by the vector C makes sense only if C is a column vector with the same number of components as the number of columns of the matrix A ( Example 2 The product A C of A = with C = is ( A C = 5 1 ( ( ( 1 + = Once the product of a matrix A and a column C is defined, we can introduce the product of two matrices A and C The idea is to think of C as a collection of columns and use the previous definition ( Ingegneria dell Autoveicolo, LeLing9 1 Geometria

2 Geometria Lingotto Definition 3 Let A be an m k matrix and C a k n matrix The product A C is the m n matrix A C := (A C 1 A C 2 A C n where A C j denotes the product of A by the column C j Therefore, the first column of the product A C is the linear combination of the columns of A obtained using the coefficients of the first column of C, the second column of A C is the linear combination of the columns of A with the coefficients of the second column of C et cetera In this way the product one thinks of A C from the point of view of its columns Thinking at A from the point of view of its rows, so A = R 1 R 2 R m, we see that the product A C can be calculated also multiplying the rows of A with the column C : A C = R 1 C R 2 C R m C This observation proves the following result R 1 R 2 Proposition 4 Let A = an m k matrix, so that the row R j R m has k elements, and suppose C = (C 1 C 2 C n is a k n matrix Then the product A C is this m n matrix: A C := where R i C j = k s=1 a isc sj From the point of view of rows, then, R 1 C 1 R 1 C 2 R 1 C n R 2 C 1 R 2 C 2 R 2 C n R m C 1 R m C 2 R m C n Ingegneria dell Autoveicolo, LeLing9 2 Geometria

3 1 Proof of the Rank theorem Geometria Lingotto A C = R 1 C R 2 C R m C In words: the first row of A C is the linear combination of the rows of C that uses the coefficients of the first row of A, the second row of A C is the linear combination of the rows of C using the coefficients of the second row of A and so on Here is a practical way to multiply A B : 1 Proof of the Rank theorem This way of looking at matrix multiplication via rows or columns allows us to prove in simple manner the Rank theorem, according to which the dimensions of the row and column spaces ρ R (A and ρ C (A are equal for any matrix A Consequently, we can define the rank ρ(a of a matrix A as ρ(a = ρ R (A = ρ C (A Proof of the Rank theorem Let A be n m and c = ρ C (A the dimension of the column space Then the columns of A are linear combinations of c columns C 1, C 2, C c Define C as the matrix with columns C 1, C 2,, C c From the column-interpretation of the product, there exists a matrix M such that A = C M, so the first column of A is the linear combination of the columns of C using the coefficients of the first column Ingegneria dell Autoveicolo, LeLing9 3 Geometria

4 1 Proof of the Rank theorem Geometria Lingotto of M, et cetera The matrix C has size n c and M has size c m Now look at the product A = C M using the rows The rows of A are linear combinations of the rows of M This implies that the row space R A of A is a subspace of the row space of R M of M, hence ρ R (A = dim(r A dim(r M = ρ R (M But M has c rows, so ρ R (A = dim(r A dim(r M = ρ R (M c = ρ C (A and hence ρ R (A ρ C (A Similarly, thinking of A using the rows, we obtain ρ C (A ρ R (A In conclusion then, ρ C (A = ρ R (A The algebra of n n matrices Let M n,n denote the set of square matrices n n We already know that M n,n is a vector space, but also that the product A B of two matrices of M n,n is still of size n n These facts guarantee we can talk about the algebra of square matrices, because an algebra is by definition a vector space where two vectors can by multiplied so that the usual rules valid for numbers are satisfied, ie the associative and distributive laws Proposition 5 Let A, B, C be matrices The following rules hold: (i (A B C = A (B C (associativity, (ii { (A + B C = A C + B C A (B + C = A B + A C (distributivity The distributive law is stated only apparently twice: in fact, the difference between multiplying numbers as opposed to matrices is that the product of matrices is not commutative Example 6 The following matrices show this phenomenon Let A = ( B = Hence 1 ( 1 A B =, whereas B A = ( 1 ( 1 and Ingegneria dell Autoveicolo, LeLing9 4 Geometria

5 2 Elementary row operations using the product of matricesgeometria Lingotto 2 Elementary row operations using the product of matrices We have seen that the first row of A B is the combination of the rows of B using the coefficients of the first row of A For instance, multiplying the first row of B by a number r gives: r 1 B 1 The elementary row operation (ERO rr i that multiplies the row R i by the number r can be defined by the matrix multiplication One simply replaces the 1 in position i i in the identity matrix with r For example, if B is 4 3 the operation 7R 3 is the product A B where A is: A = For obvious reasons such a matrix is denoted rr i instead of A, because writing rr i B readily says the result is that of using an ERO on the rows of B The ERO of swapping two rows is equally definable with the product of suitable matrices Suppose we want to exchange the first tow rows of B ; then it is enough to compute A B where: 1 1 A = 1 1 This is why, instead of A, we use the symbol R 1 2 to indicate this matrix Hence R 1 2 B is the matrix obtained swapping the first two rows of B This works for any couple of rows i, j if we multiply by R i j For example, instead of swapping rows two Ingegneria dell Autoveicolo, LeLing9 5 Geometria

6 2 Elementary row operations using the product of matricesgeometria Lingotto and five in B of size 5 6 we may simply calculate R 2 5 B where: 1 1 R 2 5 = The last ERO R i + rr j as well, that adds to the ith row r times the j th row r, can be done with a product The matrix of concern is indicated with R i+rj If B is 5 6, R is R = We can summarise all this in a theorem Theorem 7 Let A be a matrix and E the echelon matrix obtained from A by the Gauß-Jordan reduction Then there exists a matrix M such that M A = E Precisely, M is the product of matrices of the form R i+rj, R i j and rr i Matrices of type R i+rj, R i j, rr i are called elementary matrices The transposed matrix We have seen why it is important to emphasize the role of columns and rows of matrices If A is a matrix, the transposed matrix A t is the matrix obtained swapping the rows and columns of A: thus the row R i of A becomes the column C i of A t It s easy to see that if A = (a ij, then A t = a ji has the indices exchanged, for the rows become columns and the other way around 1 6 ( Example 8 The transpose of A = is A t = If A is n m then A t is m n Clearly, (A t t = A A matrix is called symmetric if A t = A, and skew-symmetric if A t = A Note that the diagonal entries of a skew-symmetric matrix are zero Ingegneria dell Autoveicolo, LeLing9 6 Geometria

7 2 Elementary row operations using the product of matricesgeometria Lingotto The most important property of transposition is the following Proposition 9 Let A, B be matrices: (A + B t = A t + B t, (ra t = ra t, (A B t = B t A t The first two ensure the transpose of a linear combination is a linear combination of transposed matrices: (c 1 A c n A n t = c 1 A t c n A t n the last one tells us that the order of factors changes after transposition A corollary of the Rank theorem states that: Proposition 1 The rank of A equals the rank of A t : ρ(a = ρ(a t We already know that the set M n,m of n m matrices is a vector space of dimension nm, as the matrices E ij with zeroes everywhere except in position (i, j, where there is 1, are nm LI matrices generating M n,m Let A be a column matrix of n elements and B a column with m Then A B t M n,m and A B = (a i b j The products e i (e j t, where (e i is the canonical basis of the columns, are precisely the matrices E ij With this it is easy to multiply A E ij = A (e i (e j t = (A e i (e j t = A i (e j t ; the product A E ij is thus the matrix whose columns are all zero except the j th one with coincides with the ith column of A Rotations An important application of A C is provided by rotations Suppose we want to rotate a figure in the plane by a counter-clockwise angle θ around the origin O : Ingegneria dell Autoveicolo, LeLing9 7 Geometria

8 2 Elementary row operations using the product of matricesgeometria Lingotto The problem is to determine the coordinates of the points A, B, C by means of the coordinates of the points A, B, C ( cos(θ sin(θ The matrix R = helps to solve the problem Suppose the coordinates of O are,, ie O is represented by the column vector We may sin(θ cos(θ ( ( ( ( a a represent A by A =,, the point B by B = and C by C = b b The solution is: A = R A B = R B C = R C Thus, in general, to rotate the column X = product R X : ( x y it suffices the calculate the Ingegneria dell Autoveicolo, LeLing9 8 Geometria

9 2 Elementary row operations using the product of matricesgeometria Lingotto ( cos(θ sin(θ R X = sin(θ cos(θ ( x y = ( cos(θx sin(θy sin(θx + cos(θy later on in the course we will show that rotating plane figures is easy using complex numbers Ingegneria dell Autoveicolo, LeLing9 9 Geometria