1. Fall 2006AE6382 Design Computing1 Vectors and Matrices Lecture Basic vector & matrix concepts Creating arrays and matrices Accessing matrix components Manipulating matrices Matrix functions Solving simultaneous equations Learning Objectives Understand the nature of matrices Understand how to manipulate matrices in Matlab

2. Fall 2006AE6382 Design Computing2 Using Matlab with Arrays and Matrices Matlab’s origins are in the early efforts to develop fast and efficient programs for handling linear equations… –Operations with arrays, vectors and matrices are needed –Only the most computationally efficient routines are used –Matlab is very “C-like” but adds a number of operators and extends its syntax to handle a range of array, vector and matrix operations –Matlab’s fundamental data structure is the array and vectors and matrices follow easily –BUT… to see some of the power of Matlab for engineering applications, we’ll have to dig a bit more deeply into some of the underlying math (no, this is not going to turn into a math class, but it’s often hard to avoid math in engineering)

3. Fall 2006AE6382 Design Computing3 Basic Concepts Scalars: magnitude only Vectors: magnitude AND direction Arrays: can be 2D or higher dimension x, mass, color, 13.451

4. Fall 2006AE6382 Design Computing4 Matlab Can Handle This… Scalars: Vectors: Arrays: >> whos Name Size Bytes Class a 1x1 8 double array density 1x1 8 double array mass 1x1 8 double array resistance 1x1 8 double array s 1x1 8 double array stress 1x1 8 double array >> force=[12.3, 5.67] force = 12.3000 5.6700 >> hvec=[1, 5, -3, 4, 0] hvec = 1 5 -3 4 0 >> coef=[1, 2; -4, 3] coef = 1 2 -4 3

5. Fall 2006AE6382 Design Computing5 Basic Array Operations Addition/subtraction: C=A+B where c ij = a ij +b ij Multiplication/division: C=A.* B where c ij = a ij *b ij Exponentiation: C=A.^ 4 where c ij = a ij 4 A = 1 2 -4 3 B = -4 1 -3 6 >> C=A+B C = -3 3 -7 9 >> C=A.*B C = -4 2 12 18 >> C=A./B C = -0.2500 2.0000 1.3333 0.5000 >> C=A.^2 C = 1 4 16 9

6. Fall 2006AE6382 Design Computing6 Notes on Array Operations Arithmetic operations on arrays are just like the same operations for scalars but they are carried out on an element-by-element basis. –the dot (.) before the operator indicates an array operator; it is needed only if the meaning cannot be automatically inferred. –when combining arrays, make sure they all have the same dimensions –applies to vectors, 2D arrays, multi-dimensional arrays >> A=[1 2 3 4 5]; >> 2.*A ans = 2 4 6 8 10 >> 2*A ans = 2 4 6 8 10 >> B=[2 4 6 8 10]; >> A.*B ans = 2 8 18 32 50 >> A*B ??? Error using ==> * Inner matrix dimensions must agree.

7. Fall 2006AE6382 Design Computing7 More Notes on Array Operations Most Matlab functions will work equally well with both scalars and arrays (of any dimension) Use brackets […] to construct arrays Use colon notation (e.g., A(:,2) or f(3:11) to index) >> A=[1 2 3 4 5]; >> sin(A) ans = 0.8415 0.9093 0.1411 -0.7568 -0.9589 >> sqrt(A) ans = 1.0000 1.4142 1.7321 2.0000 2.2361

8. Fall 2006AE6382 Design Computing8 Array Constructors Arrays are often read into Matlab from files or entered by the user… But building arrays from scratch can be tedious –Explicit: –Using Matlab array constructors: >> g(1)=1; g(2)=3; g(3)=-4 g = 1 3 -4 >> A=ones(2,3) A = 1 1 1 >> B=-3*ones(1,5) B = -3 -3 -3 -3 -3 >> C=zeros(2,3) C = 0 0 0

9. Fall 2006AE6382 Design Computing9 Let’s Build Some Arrays… >> A=3*eye(2,2) A = 3 0 0 3 >> B=diag([1 2 3 4]) B = 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 >> C=diag([1 2 1],1) C = 0 1 0 0 0 0 2 0 0 0 0 1 0 0 0 0 >> diag(A) ans = 3 What will these produce? D = magic(5) diag(D) diag(diag(D)) Z = [magic(3),zeros(3,2), -ones(3,1); 4*ones(2,4), eye(2,2)] Z(:,3)=[] mess = 10*rand(4,5) messy = 10*randn(4,5) test = 1./(3*ones(2,3)

10. Fall 2006AE6382 Design Computing10 Vectors and Matrices We’ve referred to vectors and matrices frequently… but exactly what are we talking about? –what is a matrix? –is it different from an array? ANSWER: –vectors and matrices are arrays with an “attitude” –that is, they look just like an array (and they are arrays), but they live by a very different set of rules! –Vectors: Can you explain what, if anything, results from these operations with vectors?

11. Fall 2006AE6382 Design Computing11 Why Matrices? A matrix is an array that obeys a different set of rules –addition & subtraction are same as for arrays, –but multiplication, division, etc. are DIFFERENT! –a matrix can be of any dimension but 2D square matrices are the most common by far A large and very useful area of mathematics deals with what is called “linear algebra” and matrices are an integral part of this. Many advanced computational methods in engineering make extensive use of linear algebra, and hence of matrices

12. Fall 2006AE6382 Design Computing12 A Simple Example A set of simultaneous linear algebraic equations will often arise in engineering applications How do you solve these? –Solve first for x in terms of y; substitute in second and solve for y; use this in first to find x –Use “Cramer’s Rule” –Other? Let’s try a more abstract notation: OR

13. Fall 2006AE6382 Design Computing13 A Simple Example-cont’d What do we mean by the * for this form? –Note that the column matrix, z, is multiplied times the first row of C on an element-by-element basis and the results are summed to get the first row of the answer –Ditto for the second row… –This is NOT array multiplication; it is matrix multiplication For two 2D matrices in general: NOTE: the number of columns in A must be equal to the number of rows in B (N in this example)

14. Fall 2006AE6382 Design Computing14 A Few Notes on Matrices Matlab handles matrix multiplication with the * symbol (NOTE: this is NOT array multiplication!) –From our formula we see that in general: A*B  B*A –In other words, matrix multiplication is NOT commutative Matrices behave just like arrays for addition and subtraction Matrix division is not strictly defined but a matrix inverse is available to address this situation, among others. –suppose: 3y=6 and you need to find y… –The usual approach: y=6/3=2 (division by 3) –Also useful: y=3 -1 *6=2 (multiplication by the inverse of 3) –If we don’t know how to divide, we can accomplish the same by using the notion of the inverse. Recall definition of inverse: –Turns out we know how to compute matrix inverses (but it requires a lot of computational effort)

15. Fall 2006AE6382 Design Computing15 Let’s Solve Our Problem Using Matlab >> coef=[3 -2; 1 4] coef = 3 -2 1 4 >> inv(coef) % Matlab has the inv() function ans = 0.2857 0.1429 -0.0714 0.2143 >> b=[14 -14]' b = 14 -14 >> z=inv(coef)*b z = 2 -4 >> coef*z % Let's check our answer! ans = 14 -14

16. Fall 2006AE6382 Design Computing16 Some More Notes: Using the Matlab inv() function is not always best –It can take a VERY long time for large matrices –The inverse may have poor precision for some kinds of matrices If you just want to solve the set of equations, there are much quicker and more accurate methods –Uses powerful algorithms from linear algebra –Notation is tricky because it introduces the concept of a “left” and a “right” matrix division in Matlab NOTE: C\C=1, and 1*anything=anything

17. Fall 2006AE6382 Design Computing17 Let’s Try This Out… coef = 3 -2 1 4 >> b b = 14 -14 >> zz=coef\b zz = 2.0000 -4.0000 OK, now what do you think these expressions yield? coef\eye(2,2) coef\eye(2,2)*coef

18. Fall 2006AE6382 Design Computing18 Things Can Get Weird… We usually think of the unknown (z) as a column matrix and the RHS (b) as a column matrix also In some fields, it is more useful if these are ROW matrices –One formulation can easily be converted into the other! –We can treat either formulation in Matlab First, ON YOUR OWN, prove from our multiplication formula that: Now, using this, we take the transpose of our equation: where

19. Fall 2006AE6382 Design Computing19 Let’s Try It Out in Matlab: >> coefT=coef' coefT = 3 1 -2 4 >> bT=b' bT = 14 -14 >> zT=bT*inv(coefT) zT = 2 -4 >> % ALSO WE CAN USE RIGHT DIVIDE: >> zT2=bT/coefT zT2 = 2.0000 -4.0000

20. Fall 2006AE6382 Design Computing20 Other Matlab Matrix Functions So far we’ve only scratched the surface of Matlab’s abilities to work with matrices… Matrices can contain COMPLEX numbers Some of the other matrix functions are: –det(A): determinant of the matrix –rank(A): rank of the matrix –trace(A): sum of diagonal terms –sqrtm(A): matrix square root (i.e., sqrtm(A)*sqrtm(A)=A) –norm(A): matrix norm (useful for vector magnitudes) –eig(A): eigenvalues and eigenvectors of matrix –… Keep in mind that Matlab is using some of the latest and most powerful algorithms to compute these functions.

21. Fall 2006AE6382 Design Computing21 Finally, What About Vectors? The matrix and array operations and functions can be used to manipulate vectors, but you’ll have to be careful Vector dot product: ON YOUR OWN: –Vector magnitude? –Vector cross product? >> f=[1 2]' f = 1 2 >> g=[4 -3]' g = 4 -3 >> fdotg=f'*g fdotg = -2 >> f=[1 2] f = 1 2 >> g=[4 -3] g = 4 -3 >> fdotg=f*g' fdotg = -2 >> gdotf=g*f' gdotf = -2 Column vectors Row vectors