Matrices Gaussian elimination Determinants. Graphics 2011/2012, 4th quarter. Lecture 4: matrices, determinants

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1 Lecture 4 Matrices, determinants

2 m n matrices Matrices Definitions Addition and subtraction Multiplication Transpose and inverse a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn is called an m n matrix with m rows and n columns. The a ij s are called the coefficients of the matrix, and m n is its dimension.

3 Special cases Matrices Definitions Addition and subtraction Multiplication Transpose and inverse A square matrix (for which m = n) is called a diagonal matrix if all elements a ij for which i j are zero. If all elements a ii are one, then the matrix is called an identity matrix, denoted with I m (depending on the context, the subscript m may be left out) I = If all matrix entries are zero (i.e. a ij = 0 for all i, j), then the matrix is called a zero matrix, denoted with 0.

4 Matrix addition Matrices Definitions Addition and subtraction Multiplication Transpose and inverse For an m A n A matrix A and an m B n B matrix B, we can define addition as A + B = C, with c ij = a ij + b ij for all 1 i m A, m B and 1 j n A, n B. For example: = Notice, that the dimensions of the matrices A and B have to fulfill the following conditions: m A = m B and n A = n B. Otherwise, addition is not defined.

5 Matrix subtraction Matrices Definitions Addition and subtraction Multiplication Transpose and inverse Similarly, we can define subtraction between an m A n A matrix A and an m B n B matrix B as A B = C, with c ij = a ij b ij for all 1 i m A, m B and 1 j n A, n B. For example: = Again, for the dimensions of the matrices A and B we must have m A = m B and n A = n B.

6 Multiplication with a scalar Definitions Addition and subtraction Multiplication Transpose and inverse Multiplying a matrix with a scalar is defined as follows: ca = B, with b ij = ca ij for all 1 i m A and 1 j n A. For example: = Obviously, there are no restrictions in this case (other than c being a scalar value, of course).

7 Matrix multiplication Matrices Definitions Addition and subtraction Multiplication Transpose and inverse The multiplication of two matrices with dimensions m A n A and m B n B is defined as For example: AB = C with c ij = n A k=1 a ik b kj ( ) ( ) = Again, we see that certain conditions have to be fulfilled, i.e. n A = m B. The dimensions of the resulting matrix C are m A n B.

8 Matrix multiplication Matrices Definitions Addition and subtraction Multiplication Transpose and inverse Useful notation when doing this on paper: ( )

9 Properties of matrix multiplication Definitions Addition and subtraction Multiplication Transpose and inverse Matrix multiplication is distributive over addition: A(B + C) = AB + AC (A + B)C = AC + BC and it is associative: (AB)C = A(BC) However, it is not commutative, i.e. in general, AB is not the same as BA.

10 Properties of matrix multiplication Definitions Addition and subtraction Multiplication Transpose and inverse Proof that matrix multiplication is not commutative, i.e. that in general, AB BA. Proof:

11 Definitions Addition and subtraction Multiplication Transpose and inverse Properties of matrix multiplication Proof that matrix multiplication is not commutative, i.e. that in general, AB BA. Alternative proof (proof by counterexample): Assume two matrices A = ( ) 1 2 and B = 3 4 ( ) ( ) ( ) ( ) ( )

12 Identity and zero matrix revisited Definitions Addition and subtraction Multiplication Transpose and inverse Identity matrix I m : I = With matrix multiplication we get IA = AI = A (hence the name identity matrix ).

13 Identity and zero matrix revisited Definitions Addition and subtraction Multiplication Transpose and inverse Zero matrix 0: = With matrix multiplication we get 0A = A0 = 0.

14 Transpose of a matrix Matrices Definitions Addition and subtraction Multiplication Transpose and inverse The transpose A T of an m n matrix A is an n m matrix that is obtained by interchanging the rows and columns of A, so a ij becomes a ji for all i, j: a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn a 11 a 21 a m1 A T a 12 a 22 a m2 = a 1n a 2n a mn

15 Transpose of a matrix Matrices Definitions Addition and subtraction Multiplication Transpose and inverse Example: A = ( 1 2 ) A T = For the transpose of the product of two matrices we have (AB) T = B T A T

16 Transpose of a matrix Matrices Definitions Addition and subtraction Multiplication Transpose and inverse For the transpose of the product of two matrices we have (AB) T = B T A T. Let s look at the left side first: b 11 b 1j b 1nB B = a 11 a 1nA A = a i1 a ina, AB = a ma 1 a ma n A b na 1 b na j b na n B c 11 c 1nB c ij c ma 1 c ma n B So, for AB, we get c ij = a i1b 1j + a i2b 2j + + a ina b na j for all 1 i m A and 1 j n B

17 Transpose of a matrix Matrices Definitions Addition and subtraction Multiplication Transpose and inverse Now let s look at the right side of the equation: a 11 a i1 a ma 1 A T = b 11 b na B T = b 1j b na j, B T A T = b 1nB b na n B a 1nA a ina a ma n A d 11 d 1nB d ji d ma 1 d ma n B So, for B T A T we get d ji = b 1ja i1 + b 2ja i2 + + b na ja ina for all 1 i m A and 1 j n B

18 Comment Matrices Definitions Addition and subtraction Multiplication Transpose and inverse Notice the differences in the two proofs: For AB BA, we showed that the general statement AB = BA is not true. Hence, it was enough to give one example where this general statement was not fulfilled. For (AB) T = B T A T, we needed to show that this general statement is true. Hence, it is not enough to just give one (or more) examples, but we have to prove that it is fulfilled for all possible values.

19 Inverse matrices Matrices Definitions Addition and subtraction Multiplication Transpose and inverse The inverse of a matrix A is a matrix A 1 such that AA 1 = I Only square matrices possibly have an inverse. Note that the inverse of A 1 is A, so we have AA 1 = A 1 A = I

20 The dot product revisited Definitions Addition and subtraction Multiplication Transpose and inverse If we regard (column) vectors as n 1 matrices, we see that the dot product of two vectors can be written as u v = u T v: = ( ) 4 5 = (A 1 1 matrix is simply a number, and the brackets are omitted.) Note: Remember the different vector notations, e.g. v 1 v = v 2 = (v 1, v 2, v 3 ) = (v 1 v 2 v 3 ) T v 3

21 Linear equation systems Geometric interpretation Linear equation systems (LES) The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. a m1 x 1 + a m2 x a mn x n = b m can be written as a matrix equation by Ax = b, or in full a 11 a 12 a 1n x 1 b 1 a 21 a 22 a 2n x = b 2. a m1 a m2 a mn x n b m

22 Linear equation systems Geometric interpretation LESs in graphics Suppose we want to solve the following system: x + y + 2z = 17 2x + y + z = 15 x + 2y + 3z = 26 Q: what is the geometric interpretation of this system?

23 Linear equation systems Geometric interpretation LESs in graphics Suppose we want to solve the following system: x + y + 2z 17 = 0 2x + y + z 15 = 0 x + 2y + 3z 26 = 0 Q: what is the geometric interpretation of this system?

24 Linear equation systems Geometric interpretation If an LES has a unique solution, it can be solved with Gaussian elimination. Matrices are not necessary for this, but very convenient, especially augmented matrices. The augmented matrix corresponding to a system of m linear equations is a 11 a 12 a 1n b 1 a 21 a 22 a 2n b a m1 a m2 a mn b m

25 Linear equation systems Geometric interpretation LESs in graphics Example: The previous LES: x + y + 2z 17 = 0 2x + y + z 15 = 0 x + 2y + 3z 26 = 0 And its related augmented matrix:

26 Linear equation systems Geometric interpretation Basic idea of : Apply certain operations to the matrix that do not change the solution, in order to bring the matrix into a from where we can immediately see the solution. These permitted operations in are interchanging two rows. multiplying a row with a (non-zero) constant. adding a multiple of another row to a row.

27 Linear equation systems Geometric interpretation : example Permitted operations: (1) exchange rows (2) multiply row with scalar (3) add multiple of 1 row to another

28 Linear equation systems Geometric interpretation : example Remember that the augmented matrix represents the linear equation system Ax = b, i.e is equivalent to 1x + 0y + 0z = 3 0x + 1y + 0z = 4 0x + 0y + 1z = 5 which directly gives us the solution x = 3, y = 4, and z = 5 Q: what is the geometric interpretation of this solution?

29 Linear equation systems Geometric interpretation Intersection of two planes Q: what is the geometric interpretation of this system? 1st: intersection of 2 planes

30 Linear equation systems Geometric interpretation Intersection of three planes

31 Linear equation systems Geometric interpretation Q: what about the other cases (e.g. line or no intersection)? Q: and what if an LGS can not be reduced to a triangular form? Three possible situations: We get a line 0x + 0y + 0z = b (with b 0) Or we get one line like 0x + 0y + 0z = 0 Or we get two lines like 0x + 0y + 0z = 0

32 Linear equation systems Geometric interpretation Note: In the literature you also find a slightly different procedure 1st: Put 0 s in the lower triangle... (forward step) 2nd:.. then work your way back up (backward step) Another note: can also be used to invert matrices (and you will do this in the tutorials)

33 Matrices Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants The determinant of an n n matrix is the signed volume spanned by its column vectors. The determinant det A of a matrix A is also written as A. For example, ( ) a11 a A = 12 a 21 a 22 det A = A = a 11 a 12 a 21 a 22 (a 12, a 22 ) (a 11, a 21 )

34 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants : geometric interpretation In 2D: deta is the oriented area of the parallelogram defined by the 2 vectors. det A = A = a 11 a 12 a 21 a 22 (a 12, a 22 ) In 3D: deta is the oriented area of the parallelepiped defined by the 3 vectors. (a 11, a 21 )

35 Computing determinants Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Using Laplace s expansion determinants can be computed as follows: The determinant of a matrix is the sum of the products of the elements of any row or column of the matrix with their cofactors If only we knew what cofactors are...

36 Cofactors Matrices Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Take a deep breath... The cofactor of an entry a ij in an n n matrix A is the determinant of the (n 1) (n 1) matrix A that is obtained from A by removing the i-th row and the j-th column, multiplied by 1 i+j. Right: long live recursion!

37 Cofactors Matrices Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Example: for a 4 4 matrix A, the cofactor of the entry a 13 is A = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 a 21 a 22 a 24 a c 13 = a 31 a 32 a 34 a 41 a 42 a 44 and A = a 11 a c 11 + a 12a c 12 + a 13a c 13 + a 14a c 14

38 and cofactors Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Example: = = 0 1(3 8 ( 1) (1+1) ( 1) (1+2) ) +2(4 6 ( 1) (1+2) ( 1) (2+2) ) = 0 1(24 30) + 2(21 24) = 0.

39 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Computing determinants for 3 3 matrices 3x3 matrices: a b c d e f g h i = a e f h i... = (aei + bfg + cdh) (ceg + afh + bdi) An easy way to do this on paper (Rule of Sarrus): a b c a b c d e f d e f g h i g h i

40 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Computing determinants for 2 2 matrices It also works for 2x2 matrices: a b c d = ad bc But unfortunately not for higher dimensions than 3!

41 The cross product revisited Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Let s look at another example: x y z v 1 v 2 v 3 w 1 w 2 w 3 From we get x y z x y z v 1 v 2 v 3 v 1 v 2 v 3 w 1 w 2 w 3 w 1 w 2 w 3 (v 2 w 3 v 3 w 2 ) x + (v 3 w 1 v 1 w 3 ) y + (v 1 w 2 v 2 w 1 ) z

42 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Systems of linear equations and determinants Consider our system of linear equations again: x + y + 2z = 17 2x + y + z = 15 x + 2y + 3z = 26 Such a system of n equations in n unknowns can be solved by using determinants using Cramer s rule: If we have Ax = b, then x i = Ai A where A i is obtained from A by replacing the i-th column with b.

43 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Systems of linear equations and determinants So for our system x y = z 26 we have x = y = z =

44 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Systems of linear equations and determinants x i = Ai A. Why? In 2D: a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 From the image we see: x 1 a 1 a 2 = b a 2 (because shearing a parallelogram doesn t change its volume) and x 1 a 1 a 2 = b a 2 (because scaling one side of a parallelogram changes its volume by the same factor)

45 Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants Systems of linear equations and determinants x i = Ai A. Why? In 2D: a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 From the image we see: x 1 a 1 a 2 = b a 2 (because shearing a parallelogram doesn t change its volume) and x 1 a 1 a 2 = b a 2 (because scaling one side of a parallelogram changes its volume by the same factor)

46 and inverse matrices Signed volume Computing determinants via cofactors Computing determinants differently Solving LESs with determinants Inverting matrices with determinants can also be used to compute the inverse A 1 of an invertible matrix A: A 1 = à A where à is the adjoint of A, which is the transpose of the cofactor matrix of A. The cofactor matrix of A is obtained from A by replacing every entry a ij by its cofactor a c ij.

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