Matlab(tm) Quick Reference

```                    A BRIEF LIST OF MATLAB COMMANDS

Some Basic Commands (Note command syntax is case-sensitive!)

quit      quits matlab, returning you to the operating system.
exit      same as quit.
who       lists all of the variables in your matlab workspace.
whos      list the variables and describes their matrix size.
clear     deletes all matrices from active workspace.
clear x   deletes the matrix  x  from active workspace.
...       the ellipsis defining a line continuation is three
successive periods.
save      saves all the matrices defined in the current
session into the file, matlab.mat.
save filename  saves the contents of workspace into
filename.mat
save filename x y z
saves the matrices x, y and z into the file titled
filename.mat.
workspace; the file can be a binary (.mat) file
or an ASCII file.
!         the ! preceding any unix command causes the unix
command to be executed from matlab.

Commands involving functions

inline    define a function, e.g.  f = inline('x^2')
fplot     generate the graph of a function, e.g. fplot(f,[a,b])
plots f (a string defn) over the interval [a,b]
fzero     find the zero of a function, e.g.  fzero(f,x0),
computes zero of f near x0 or if x0 a vector then
in the interval [x0_1, x0_2].

Commands Useful in Plotting.

plot(x,y) creates an Cartesian plot of the vectors x & y.
plot(y)   creates a plot of y vs. the numerical values of the
elements in the y-vector.
semilogx(x,y) plots log(x) vs y.
semilogy(x,y)  plots x vs log(y)
loglog(x,y)    plots log(x) vs log(y).
grid      creates a grid on the graphics plot.
title('text')  places a title at top of graphics plot.
xlabel('text') writes 'text' beneath the x-axis of a plot.
ylabel('text') writes 'text' beside the y-axis of a plot.
text(x,y,'text')    writes 'text' at the location (x,y) .
text(x,y,'text','sc') writes 'text' at point x,y assuming
lower left corner is (0,0) and upper
right corner is (1,1).
gtext('text') writes text according to placement of mouse
hold on   maintains the current plot in the graphics window
while executing subsequent plotting     commands.
hold off  turns OFF the 'hold on' option.
polar(theta,r) creates a polar plot of the vectors r & theta
bar(x)    creates a bar graph of the vector x. (Note also
the command stairs(y).)

bar(x,y)  creates a bar-graph of the elements of the vector y,
locating the bars according to the vector elements
of 'x'. (Note also the command stairs(x,y).)
hist(x)   creates a histogram. This differs from the bargraph
in that frequency is plotted on the vertical axis.
mesh(z)   creates a surface in xyz space where z is a matrix
of the values of the function z(x,y). z can be
interpreted to be the height of the surface above
some xy reference plane.
surf(z)   similar to mesh(z), only surface elements depict
the surface rather than a mesh grid.
contour(z)     draws a contour map in xy space of the function
or surface z.
meshc(z)  draws the surface z with a contour plot beneath it.
meshgrid  [X,Y]=meshgrid(x,y) transforms the domain specified
by vectors x and y into arrays X and Y that can be
used in evaluating functions for 3D mesh/surf plots.
print     sends the contents of graphics window to printer.
print filename -dps      writes the contents of current
graphics to 'filename' in postscript format.

Equation Fitting

polyfit(x,y,n) returns the coefficients of the n-degree
polynomial for the vectors x and y. n must be at least 1
larger than the length of the vectors x and y. If n+1 =
length(x) the result is an interpolating polynomial. If
n+1 > length(x) the result is a least-squares polynomial
fit. The coefficients are stored in order with that of
the highest order term first and the lowest order last.
polyval(c,x)   calculates the values of the polynomial whose
coefficients are stored in  c, calculating for every
value of the vector x.

Data Analysis Commands

max(x)    returns the maximum value of the elements in a
vector or if x is a matrix, returns a row vector
whose elements are the maximum values from each
respective column of the matrix.
min (x)   returns the minimum of x (see max(x) for details).
mean(x)   returns the mean value of the elements of a vector
or if x is a matrix, returns a row vector whose
elements are the mean value of the elements from
each column of the matrix.
median(x) same as mean(x), only returns the median value.
sum(x)    returns the sum of the elements of a vector or if x
is a matrix, returns the sum of the elements from
each respective column of the matrix.
prod(x)   same as sum(x), only returns the product of
elements.
std(x)    returns the standard deviation of the elements of a
vector or if x is a matrix, a row vector whose
elements are the standard deviations of each
column of the matrix.
sort(x)   sorts the values in the vector x or the columns of
a matrix and places them in ascending order. Note
that this command will destroy any association that
may exist between the elements in a row of matrix x.
hist(x)   plots a histogram of the elements of vector, x. Ten
bins are scaled based on the max and min values.
hist(x,n) plots a histogram with 'n' bins scaled between the
max and min values of the elements.
hist((x(:,2))  plots a histogram of the elements of the 2nd
column from the matrix x.
fliplr(x) reverses the order of a vector. If x is a matrix,
this reverse the order of the columns in the matrix.
flipud(x) reverses the order of a matrix in the sense of
exchanging or reversing the order of the matrix
rows. This will not reverse a row vector!
reshape(A,m,n)      reshapes the matrix A into an mxn matrix
from element (1,1) working column-wise.

SPECIAL MATRICES

zeros(n)  creates an nxn matrix whose elements are zero.
zeros(m,n)   creates a  m-row, n-column matrix of zeros.
ones(n)   creates a n x n square matrix whose elements are 1's
ones(m,n)' creates a mxn matrix whose elements are 1's.
ones(A)   creates an m x n matrix of 1's, where m and n are
based on the size of an existing matrix, A.
zeros(A)  creates an mxn matrix of 0's, where m and n are
based on the size of the existing matrix, A.
eye(n)    creates the nxn identity matrix with 1's on the
diagonal.

Miscellaneous Commands

length(x) returns the number elements in a vector.
size(x)   returns the size m(rows) and n(columns) of matrix x.
rand      returns a random number between 0 and 1.
randn     returns a random number selected from a normal
distribution with a mean of 0 and variance of 1.
rand(A)   returns a matrix of size A of random numbers.

ALGEBRAIC OPERATIONS IN MATLAB

Scalar Calculations.
-         subtraction
*         multiplication
/         right division (a/b means a ÷ b)
\         left division  (a\b means b ÷ a)
^         exponentiation

The precedence or order of the calculations included in a single
line of code follows the below order:
Precedence                Operation
1         parentheses
2         exponentiation, left to right
3         multiplication and division, left  right
4         addition and subtraction, left  right

MATRIX ALGEBRA
In matrix multiplication, the elements of the product, C, of
two matrices A*B is calculated from

Cij = ‘ (Aik * Bkj)   {summation over the double index k}

To form this sum, the number of columns of the first or left matrix
(A) must be equal to the number of rows in the second or right
matrix (B). The resulting product, matrix C, has an order for which
the number of rows equals the number of rows of the first (left)
matrix (A) and the product (C) has a number of columns equal to the
number of columns in the second (right) matrix (B). It is clear
that A*B IS NOT NECESSARILY EQUAL TO B*A!

The PRODUCT OF A SCALAR AND A MATRIX is a matrix in which
every element of the matrix has been multiplied by the scalar.

ARRAY PRODUCTS

Sometimes it is desired to simply multiply or divide each
element of an matrix by the corresponding element of another
matrix. These are called 'array operations" in 'matlab'. Array or
element-by-element operations are executed when the operator is
preceded by a '.' (period). Thus

a .* b    multiplies each element of a by the respective
element of b
a ./ b    divides each element of a by the respective element
of b
a .\ b    divides each element of b by the respective element
of a
a .^ b    raise each element of a by the respective b element

TRANSPOSE OF A MATRIX

x'        The transpose of a matrix is obtained by
interchanging the rows and columns. The 'matlab'
operator that creates the transpose is the single
quotation mark, '.

INNER PRODUCT OF TWO VECTORS

The inner product of two row vectors G1 and G2 is G1*G2'.
The inner product of two column vectors H and J is H'*J.

OUTER PRODUCT OF TWO VECTORS

If two row vectors exist, G1 and G2, the outer product is
simply

G1' * G2       {Note G1' is nx1 and G2 is 1xn}

and the result is a square matrix in contrast to the scalar result
for the inner product. DON'T CONFUSE THE OUTER PRODUCT WITH THE
VECTOR PRODUCT IN MECHANICS! If the two vectors are column vectors,
the outer product must be formed by the product of one vector times
the transpose of the second!

SOLUTION TO SIMULTANEOUS EQUATIONS

Using the Matrix Inverse

inv(a)    returns the inverse of the matrix  a.
If ax=b is a matrix equation and  a  is the
coefficient matrix, the solution  x  is  x=inv(a)*b.

Using Back Substitution

a\b       returns a column vector solution for the matrix
equation  ax=b where a is a coefficient matrix.
b/a       returns a row vector solution for the matrix
equation  xa=b where a is a coefficient matrix.
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