Matrix-Algebra

# Matrix-Algebra - Matrices Operations Linear Systems Special...

This preview shows pages 1–5. Sign up to view the full content.

Matrices Operations Linear Systems Special Matrices Matrix Algebra Dhavide Aruliah Faculty of Science MATH 2070/2072 Matrices Operations Linear Systems Special Matrices Outline Matrices, Vectors, & Scalars Operations with Matrices Systems of Linear Equations Special Matrix Families

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Matrices Operations Linear Systems Special Matrices Matrices Matrix A R m × n is rectangular array of numbers A = a 1,1 a 1,2 · · · a 1, n - 1 a 1, n a 2,1 a 2,2 · · · a 2, n - 1 a 2, n . . . . . . . . . . . . a m - 1,1 a m - 1,2 · · · a m - 1, n - 1 a m - 1, n a m ,1 a m ,2 · · · a m , n - 1 a m , n Numbers a i , j = elements of A = entries of A First index i of element a i , j = row index Second index j of element a i , j = column index Matrices Operations Linear Systems Special Matrices Vectors Special case of a “skinny” matrix n -vector is matrix x of dimension n × 1 x = x 1 x 2 . . . x n - 1 x n or x T = ( x 1 x 2 · · · x n - 1 x n ) Elements x i = components of x Convention: vectors generically column vectors assume x R n means x R n × 1 Scalars are vectors of length 1/matrices of dimension 1 × 1
Matrices Operations Linear Systems Special Matrices Vectors and Matrices in M ATLAB Columns separated by , (optional), rows by ; Indexing: A 2,3 A ( 2 , 3 ) Colon indexing: A : ,2 means 2 nd column of A A 3,: means 3 rd row of A A 1: 3,2: 5 means 1 st –3 rd rows, 2 nd –5 th columns of A Dimensions of matrix A returned by size >> A = [ 1 2 3; 4 5 6 ]; >> size(A) >> size(A,1) >> size(A,2) Matrices Operations Linear Systems Special Matrices Matrix Addition If A R m × n , B R m × n , C = A + B R m × n has elements c i , j = a i , j + b i , j ( 1 i m , 1 j n ) Matrix addition in M ATLAB uses operator + >> A = [ 1 2 3; 4 5 6]; >> B = [ 7 8 9; 10 11 12 ]; >> C = A + B Matrices must be conformable (same shape) for addition >> a = [1,2,3]; >> b = [4;5;6]; >> a+b % Common mistake!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Matrices Operations Linear Systems Special Matrices Scalar Multiplication If μ R and A R m × n , C = μ A R m × n has elements c i , j = μ a i , j ( 1 i m , 1 j n ) Scalar multiplication in M ATLAB uses operator * >> A = [ 1 2 3; 4 5 6 ]; >> mu = 0.5; >> C = mu * A Matrices Operations Linear Systems Special Matrices Matrix Multiplication If A R m × s , B R s × n , matrix product C = AB R m × n has elements c i , j = s k = 1 a i , k b k , j ( 1 i m , 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern