Matlab Dictionary for CSC 412

General

• use help to get help on any function, e.g. help transpose;

Arithmetic

• basic arithmetic: +, -, *, /.
• array division: ./
• power: ^, inverse: ^(-1), e.g. 3^(-1) for 1/3.
• special functions: sin(x), etc.
• ranges via the colon operator: :, a:b, a:s:b, e.g. 1:10 and 1:2:10
• setting a variable: =
• comparison: ==, <, >, ~= (for not equal)

Functions and Plotting

• declaring variable(s): syms
• plotting a function f: plot(x, f(x))
• solving equations: solve()
• plotv(x): plot two-dimensional vector x starting at origin
• scatter(x,y): plot points (x(i), y(i)); scatter(x,y,1) will draw points (rather than disks)
• ezplot(f(x), [a b]): plot function f in range [a,b]; x needs to be symbolic variable, so use syms x first
• special functions:
  • max(v): maximum value in a vector, need to use twice for matrix, since it's a vector of vectors, similarly min(v)
  • abs(x): absolute value, mathematicans would write |x|
  • sin(x), tan(x), etc.: trigonometric functions
  • exp(x): exponential function
  • log(x): natural logarithm, also log2, log10
  • heaviside(x): Heaviside function
  • ceil(x), floor(x): floor and ceiling functions

Calculus

• limit(f(x),x,a): lim of f(x) as x tends to a; x needs to be declared a symbolic variable usins syms x, also 'left' and 'right' version
• diff(f(x)): derivative of f, make sure x is a symblic variable, diff(f(x), k) will give you the kth derivative

Matrix Operations and Functions

• Creating vectors:
  • v = [3 1 5];
  • randperm(n): random permutation of 1:n
• Creating Matrices:
  • M = [1 2 3; 4 5 6];
  • randi(max, n, m) function: create random matrix of dimension n x m with maximum entry max
  • eye(n): identify matrix of dimension n x n (also eye(n,m) and other forms)
  • diag(v): create diagonal matrix with diagonal entries given by vector v
  • zeros(n), zeros(n,m): n x m all zero matrix
• Matrix arithmetic: +, -, *, also / and ^ (in particular, ^(-1) for inverse)
• Matrix indexing:
  • A(i,j), A(i,:), A(:,j), etc. (this uses the colon operator :)
  • in general: A(u,v), where u and v are vectors, e.g. A([1 3]; [1 3 5]) results in a 2x3 submatrix of A
• Vector operations and functions
  • norm(v): Eucliden norm of v (other norms are available, also for matrices)
  • dot(u,v): dot/inner product of vectors
• Matrix functions:
  • size(A): dimension of matrix A, returns vector
  • trace(A): trace of matrix A
  • det(A): determinant of A
  • rank(A): rank of A (dimension of column space of A = dimension of row space of A)
  • transpose(A): tranpose of A, also A.'
  • isequal(A,B): are matrices A and B equal? A == B compares column vectors
  • charpoly(A): characteristic polynomial of A, that is, det(t*I-A)
  • norm(A, 'fro'): Frobenius norm of A: square root of sums of squares of entries in A.
• Linear Systems, Matrix decompositions, and other Matrix algorithms:
  • rref(A): reduced row echelon form
  • linsolve(A,b): solve linear system Ax = b using LU decomposition with partial pivoting (for square matrix), otherwise QR factorization
  • lu(A): LU decomposition with pivoting, {L U] = lu(A) returns L, U so that A = L*U, but L is permuted lower triangular; [L U P] = lu(A) will return L, U, and P so that P*A = L*U.
  • null(A): orthonormal basis for null space of A, i.e. {x: Ax = 0}
  • orth(A): orthonormal basis for columnspace of A, use orth(A, 'r') to get a rational basis
  • qr(A,0): economy QR factorization, i.e. A = QR with Q orthonormal (Q'*Q = I), R upper triangular; if A = mxn, then Q = mxn, and R = nxn. In the full factorization, [Q R] = qr(A), Q has dimension mxm.
  • eig(A): eigenvalues of A, if used as [P D] = eig(A), P contains eigenvectors and D is diagonal matrix of corresponding eigenvalues; in particular, A = P*D*P^(-1).
  • svd(A): singular value decomposition; use as [U D V] = svd(A), then A = PDV'