Understanding Matrices in Matlab: Basics and Operations

1. Matrices

2. • An m * n matrix is a two-dimensional array of numbers consisting of m rows and n columns. • Special cases are a column vector (n = 1) and a row vector • (m = 1). • Matrices are fundamental to Matlab and even if you are not intending to use Matlab for linear algebra computations you need to become familiar with matrix generation and manipulation

3. Matrices  A Matrix array is two-dimensional, having both multiple rows and multiple columns, similar to vector arrays:  it begins with [, and end with ]  spaces or commas are used to separate elements in a row  semicolon or enter is used to separate rows. •Example: •>> f = [ 1 2 3; 4 5 6] f = 1 2 3 4 5 6 A is an m x n matrix. the main diagonal

4. Matrices can be built explicitly using the square bracket notation. • For example, a 3-by-3 matrix comprising the rst 9 primes can be set up with the command A = [2 3 5 7 11 13 17 19 23] • The end of a row can be specified by a semicolon instead of a carriage return, so a more compact command with the same effect is “Semi-colons separate Rows” • A = [2 3 5; 7 11 13; 17 19 23] • Within a row, elements can be separated by spaces or by commas \Commas separate Columns" .

5. Matrix Addressing -- matrixname(row, column) -- colon may be used in place of a row or column reference to select the entire row or column. Example: recall: f =  >> f(2,3) 1 2 3 4 5 6 h = 2 4 6 1 3 5 ans = 6 >> h(:,1) ans = 2 1

6. Matrices (con’t…) more commands B = A’ eye(n)  returns an n x n identity matrix eye(m,n)  returns an m x n matrix with ones on the main diagonal and zeros elsewhere. C = A + B C = A – B B = A, where  is a scalar. C = A*B B = inv(A), A must be a square matrix in this case. rank (A)  returns the rank of the matrix A. B = A.^2  squares each element in the matrix C = A * A  computes A*A, and A must be a square matrix. det (A), and A must be a square matrix. A, B, C are matrices, and m, n,  are scalars. Transpose Identity Matrix Addition and subtraction Scalar Multiplication Matrix Multiplication Matrix Inverse Matrix Powers Determinant

7. eye eye Identity matrix I = eye returns the scalar, 1. I = eye(n) returns an n-by-n identity matrix with ones on the main diagonal and zeros elsewhere. I = eye(4) I = 4×4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 I = eye(n,m) returns an n-by-m matrix with ones on the main diagonal and zeros elsewhere. I = eye(2,3) 1 0 0 0 1 0

8. eye eye Identity matrix • The values of the vector in eye([2,3]) defines the size of the vector. • For example, eye([2,3]) returns a 2-by-3 array with ones on the main diagonal and zeros elsewhere. sz = [3,1]; I = eye(sz) I = 3×1 1 0 0

9. ones ones Create array of all ones Syntax X = ones %returns the scalar 1 X = ones(n) X=ones([2*3]) % returns an n*n matrix of ones Returns 2*3 array of ones

10. zeros zeros Create array of all zeros Syntax X = zeros %returns the scalar 0 X = zeros(n) X = zeros(sz) % returns an n*n matrix of zeros % returns an array of zeros where the size vector, sz, defines size (x) For example A = [1 4; 2 5; 3 6]; sz = size(A); X = zeros(sz) X = 3×2 0 0 0 0 0 0