§1 Basic ConceptsSome ShortcutsSome TipsNumerical MatricesVariablesArithmetic Expressions and Mathematical Functions§2 Matrix OperationsBasic OperationsMatrix Functions and OperatorsMatrix Creation

§1 Basic Concepts

Some Shortcuts


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1
exit,quit % quit
2
Ctrl-C % interruption
3
Ctrl-l % moving one word to the left
4
Ctrl-r % .......................right
5
Ctrl-a % moving to the beginning of the line
6
Ctrl-e % ..............end..................
7
Ctrl-k % deleting from here to the end of the line

Some Tips

You are allowed to define more than one variable on the same line, provided they are separated by ';'.
You can input '...' at the end of the line in order to continue typing on the next line.
In MATLAB, The elements of the matrix are indexed by column first and then row.

Date Formats

Numbers	Forms (BACK)
0	22-Apr-2020 02:14:00
1	22-Apr-2020
2	04/22/20
3	Apr
4	A (April)
5	4 (April)
6	04/22
7	22
8	Thu (Thursday)
9	T (Thursday)
10	2020
11	20 (2020)
12	Apr20
13	02:14:00
14	02:14:00 AM
15	02:14
16	02:14 AM
17	$2_{nd}$ quarters of 1 hour-2020)
18	Q2

Output Formats

Normally, integer or short float with 4 decimal places

Format Changes


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1
>> format
2
>> p=1/3-1
3
4
p =
5
6
   -0.6667
7
8
>> format short;p
9
10
p =
11
12
   -0.6667
13
14
>> format long;p
15
16
p =
17
18
  -0.666666666666667
19
20
>> format short e;p
21
22
p =
23
24
  -6.6667e-01
25
26
>> format long e;p
27
28
p =
29
30
    -6.666666666666667e-01
31
32
>> format hex;p
33
34
p =
35
36
   bfe5555555555556
37
38
>> format +;p
39
40
p =
41
42
-
43
44
>> format bank;p
45
46
p =
47
48
         -0.67
49
50
>> format rat;p
51
52
p =
53
54
      -2/3
55
      
56
>> format compact;p
57
p =
58
      -2/3

Help Commands

Numerical Matrices

2-dim Matrices

row: separated by ' ' or ','
column: separated by ';' or Enter


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% e.g.
2
>> A=[1,2,3;4,5,6;]
3
% or
4
>> A=[1 2 3
5
4 5 6]
6
7
A =
8
9
     1     2     3
10
     4     5     6


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% You can also assign to the elements of a matrix one by one.
2
>> B ( 1,1 ) = 1;
3
B ( 1,2 ) = 7;
4
B ( 2,1 ) =-5;
5
B ( 2,2 ) = 0
6
7
B =
8
9
     1     7
10
    -5     0

Multi-dim Matrices


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% e.g.1 constructed from 2 known 2-dim matrices
2
>> A=[1 2 3
3
4 5 6];
4
>> B=[11 12 13
5
14 15 16];
6
>> C(:,:,1)=A;
7
>> C(:,:,2)=B
8
9
C(:,:,1) =
10
11
     1     2     3
12
     4     5     6
13
14
15
C(:,:,2) =
16
17
    11    12    13
18
    14    15    16
19
% e.g.2 change one element of the above C
20
>> C(1,1,1)=100
21
22
C(:,:,1) =
23
24
   100     2     3
25
     4     5     6
26
27
28
C(:,:,2) =
29
30
    11    12    13
31
    14    15    16

Dimensions and Lengths

size(#)


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% size(Matrix_name): giving the length of every dim
2
>> size(A)
3
4
ans =
5
6
     2     3
7
% size(Matrix_name,Dim_number): giving the length of the specific dim
8
>> size(A,2)
9
10
ans =
11
12
     3
13
% size(Row_vector_name/Column_vector_name):returning 1 & the length of the row vector / the length of the column vector & 1
14
>> X=[1 2 3 4 5]
15
16
X =
17
18
     1     2     3     4     5
19
20
>> size(X)
21
22
ans =
23
24
     1     5

length(#)


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% length(Vector_name):giving the length of the vector
2
>> length(X)
3
4
ans =
5
6
     5
7
% length(Matrix_name):giving the maximum of the length of each dimension of the matrix
8
>> length(C)
9
10
ans =
11
12
     3

ndims(#)


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% ndims(Matrix_name): giving the dimension of the matrix
2
>> ndims(C)
3
4
ans =
5
6
     3

Conversion between Subscripts and indexes

sub2ind


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% sub2ind(size,Sub_No): The given subscripts of elements will be converted into homologous indexes.
2
>> ind=sub2ind(size(A),[1,2;1,2],[1,1;3,2])
3
4
ind =
5
6
     1     2
7
     5     4

ind2sub


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% ind2sub(size,Ind_No): This is the inverse process of the mentioned above.
2
>> [I,J]=ind2sub(size(A),ind)
3
4
I =
5
6
     1     2
7
     1     2
8
9
10
J =
11
12
     1     1
13
     3     2

Variables

Some Basic Rules
1. In MATLAB, variables are case sensitive and the predefined are usually lowercase.
2. 'A~Z', 'a~z' and '_' can be used to name a variable, but the first char of the name must be a letter.
Simple Data Types
Types Explanations
double Double-precision floating-point format/64-digit
char 16-digit
sparse To storage sparse matrix
unit8 8-digit unsigned int

Types	Explanations
`double`	Double-precision floating-point format/64-digit
`char`	16-digit
`sparse`	To storage sparse matrix
`unit8`	8-digit unsigned int

Logical Functions

Functions	Explanations
`iscell(x)`	If true, return 1; Else return 0/Cell matrix
`isfield(x)`	Ditto
`isfinite(x)`	Return the same vector as x
`islogical(x)`	$1_{st}$ /Logical vector
`isnumeric(x)`	Ditto/Numeric vector
`isstr(x)`	Ditto/String
`isstruct(x)`	Ditto/Structure
`isobject(x)`	Ditto/Object
`logical(x)`	Return a logical vector that can be used

Predefined Variables

Variables	Explanations
`ans`	assigned the value of the last expression, which doesn't have a name
`eps`	the diff between 1 and the closest representative floating-point number You can set a new value, but it can't restore by the command 'clear'.
`realmax` `realmin`	the biggest/smallest float-point number
`pi`	$\pi$
`inf`	$\frac{1}{0}$ When 0 becomes the divisor, it will return 'inf' instead of interruption.
`NaN`	$\frac{inf}{inf}$
`i` `j`	$\sqrt{-1}$ You can set new values for them and restore by the command 'clear'.

View Variables

commands	Explanations
`who`	return all the defined variables
`who global`	return all the defined global variables
`who a*`	return all the defined variables which start with 'a'
`whos` `whos global`	show more details than `who`/`who global`
`exist(namestr)`	return diff values according to the definitions of variables in 'namestr'
`inmem`	return a cell vector, which includes functions and M files in memory so far
`workspace`	open 'workspace'

Delete Variables

commands	Explanations
`clear`	delete all the defined variables and restore all the predefined except `eps`
`clear name1 name2 ...`	delete the specific variables
`clear a*`	delete all the defined variables which start with 'a'

2 equivalent descriptions:
command variable
command('variable')

Arithmetic Expressions and Mathematical Functions

Arithmetic Operators
Priority Operators
H ^
M * / \
L + -
Mathematical Functions
Functions Explanations
abs(x) absolute value
sign(x) sign function
sqrt(x) $\sqrt{x}$
pow2(x,f) $x\times 2^f$
exp(x) $e^x$
log(x) $\ln{x}$
log10(x) $\log_{10}{x}$
sin(x)... trigonometric functions
The x in trigonometric functions must be in radians.

Priority	Operators
H	^
M	* / \
L	+ -

Functions	Explanations
`abs(x)`	absolute value
`sign(x)`	sign function
`sqrt(x)`	$\sqrt{x}$
`pow2(x,f)`	$x\times 2^f$
`exp(x)`	$e^x$
`log(x)`	$\ln{x}$
`log10(x)`	$\log_{10}{x}$
`sin(x)...`	trigonometric functions

Rounding and Relevant Commands

Commands	Explanations
`round(x)`	return the integer closest to x
`fix(x)`	return the integer closest to x towards 0
`floor(x)` `ceil(x)`	round down/up
`rem(x,y)`	return the remainder of x/y
`gcd(x,y)`	GCD
`[a,b,c]=gcd(x,y)`	$a=x\cdot b+y\cdot c$
`lcm(x,y)`	LCM
`[t,n]=rat(x)`	$\frac{t}{n}\in\R$ $10^{-6}\ \ \ \ \ (t,n\in\Z)$
`[t,n]=rat(x,tol)`	... less than 'tol'
`rat(x)` `rat(x,tol)`	return continued fraction form of x with a relative error

Functions on Complex Numbers

Functions	Explanations
`real(z)` `imag(z)`	return the real/imaginary part of z
`abs(z)`	absolute value
`conj(z)`	return the complex conjugate of z
`angle(z)`	$(-\pi,\pi)$
`unwrap(v)` `unwrap(v,k)`	$\pi$ $k\pi$
`cplxpair(v)`	return complex conjugate pairs


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% e.g. cplxpair
2
>> v
3
4
v =
5
6
   1.0000 + 2.0000i   2.0000 + 3.0000i  -1.0000 - 3.0000i   1.0000 - 2.0000i  -1.0000 + 3.0000i   2.0000 - 3.0000i   2.0000 + 0.0000i
7
8
>> cplxpair(v)
9
10
ans =
11
12
  -1.0000 - 3.0000i  -1.0000 + 3.0000i   1.0000 - 2.0000i   1.0000 + 2.0000i   2.0000 - 3.0000i   2.0000 + 3.0000i   2.0000 + 0.0000i

Coordinate Conversions

Conversions	Explanations
`[theta,r]=cart2pol(x,y)`	convert Cartesian coordinate system into polar coordinate system
`[x,y]=pol2cart(theta,r)`	convert polar coordinate system into Cartesian coordinate system
`[alpha,theta,r]=cart2sph(x,y,z)`	convert Cartesian coordinate system into spherical coordinate system
`[x,y,z]=sph2cart(alha,theta,r)`	converse process of the above

~~Flops and~~ Time Management

~~flops~~
1. ~~$+ \ -$~~
  1. ~~real: 1 operation~~
  2. ~~complex: 2 operations~~
2. ~~$\times\ \div$~~
  1. ~~real: 1 operation~~
  2. ~~complex: 6 operations~~
3. ~~Elementary Functions~~
  1. ~~real: 1 operation~~
  2. ~~complex: more than 1~~
4. ~~flops(0) reset to 0~~

Time Management

Commands	Explanations
`tic`	start a timer
`toc`	read the time of the timer started by `tic`
`clock` `fix(clock)`	return the date and time (scientific notation)
`etime(t1,t2)`	return the seconds between t1 and t2
`eputime`	return the seconds of CPU since startup of MATLAB


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% e.g.
2
>> tic
3
>> toc
4
历时 2.159932 秒。
5
>> clock
6
7
ans =
8
9
   1.0e+03 *
10
11
    2.0210    0.0010    0.0230    0.0170    0.0360    0.0366
12
>> fix(clock)
13
14
ans =
15
16
        2021           1          23          17          35           9
17
>> etime([2020 1 23 0 0 0],[2021 1 23 0 0 0])
18
19
ans =
20
21
   -31622400
22
>> cputime
23
24
ans =
25
26
  111.3125

Commands	Explanations
`date`	date-month-year
`calendar(yyyy,mm)`	return the appointed calendar
`datenum(yyyy,mm,dd)`	$1_{st}$ day: 0000-01-01
`datestr(d,form)`	return the date denoted by form
`datetick(axis,form)`	write data on the coordinate axis in the figure
`datevec(d)`	return `[yyyy mm dd ho mi se]`
`enomday(yyyy,mm)`	the number of days
`now`	similar to `datenum`
`[daynr dayname]=weekday(day)`	return the day of the week (1 is Sunday)

§2 Matrix Operations

Basic Operations

$+\ -$


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% one tip
2
>> A
3
A =
4
     1     2
5
     3     4
6
>> A+100
7
ans =
8
   101   102
9
   103   104

Multiplication

$*$ 2-dim only

$\cdot$


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% dot product
2
>> x=[1 2 3 4 5];y=[1;2;3;4;5];
3
>> x*y
4
ans =
5
    55
6
>> y*x
7
ans =
8
     1     2     3     4     5
9
     2     4     6     8    10
10
     3     6     9    12    15
11
     4     8    12    16    20
12
     5    10    15    20    25
13
>> dot(x,y)
14
ans =
15
    55
16
>> dot(y,x)
17
ans =
18
    55
19
>> A=[1 2 3
20
4 5 6];...
21
B=[1 0 0
22
0 0 1];
23
>> dot(A,B) % namely dot(A,B,1)
24
ans =
25
     1     0     6
26
>> dot(A,B,2)
27
ans =
28
     1
29
     6

$\times$ similar as dot product
Convolution
Tensor Product

$\rightarrow$ inverse matrix: inv(Matrix_name)
Conjugate and Transpose
1. Conjugate: A.' or conj(A)
2. Conjugate and Transpose: A'

Exponentiation


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>> A
2
A =
3
     1     2
4
     2     1
5
>> 2^A
6
ans =
7
    4.2500    3.7500
8
    3.7500    4.2500
9
>> A^2
10
ans =
11
     5     4
12
     4     5

Element Operations + - .* ./ .\ .^

Matrix Functions and Operators

Matrix Functions
Functions
expm(Matrix_name)
logm(Matrix_name)
sqrtm(Matrix_name)
Note the difference between expm and exp etc.
Relational Operators < <= > >= == ~= The elements of the result matrix are 0(false) or 1(true).

Functions
`expm(Matrix_name)`
`logm(Matrix_name)`
`sqrtm(Matrix_name)`

Logical Operators & | ~ xor


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% e.g.
2
>> A=[1 2 0];B=[0 2 3];
3
>> A & B
4
ans =
5
  1×3 logical 数组
6
   0   1   0
7
>> A | B
8
ans =
9
  1×3 logical 数组
10
   1   1   1
11
>> ~ A
12
ans =
13
  1×3 logical 数组
14
   0   0   1
15
>> xor(A,B)
16
ans =
17
  1×3 logical 数组
18
   1   0   1
19
>> C=1;
20
>> A & C
21
ans =
22
  1×3 logical 数组
23
   1   1   0

Logical Functions

find to give the values and positions of adequate elements of the matrix or vector


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1
% e.g.1 finding nonzero elements
2
>> x=[1 2 -1 0 0 3 0 4];
3
>> find(x)
4
ans =
5
     1     2     3     6     8
6
>> A=[1 2 0
7
2 0 0
8
4 0 3];
9
>> find(A)
10
ans =
11
     1
12
     2
13
     3
14
     4
15
     9
16
>> [u,v]=find(A)
17
u =
18
     1
19
     2
20
     3
21
     1
22
     3
23
v =
24
     1
25
     1
26
     1
27
     2
28
     3
29
>> [u,v,b]=find(A)
30
u =
31
     1
32
     2
33
     3
34
     1
35
     3
36
v =
37
     1
38
     1
39
     1
40
     2
41
     3
42
b =
43
     1
44
     2
45
     4
46
     2
47
     3
48
% e.g.2 used with relational operators
49
>> find(x>1)
50
ans =
51
     2     6     8
52
>> x(ans)
53
ans =
54
     2     3     4
55
>> length(ans)
56
ans =
57
     3

any all return Boolean values


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1
%
2
>> x=[1;0;2];y=[1;1;2];z=[0;0;0];
3
>> any([x y z])
4
ans =
5
  1×3 logical 数组
6
   1   1   0
7
>> all([x,y,z])
8
ans =
9
  1×3 logical 数组
10
   0   1   0
11
>> any(any([x y z]))
12
ans =
13
  logical
14
   1
15
>> all(all([x y z]))
16
ans =
17
  logical
18
   0
19
% with relational operators
20
>> all(y>0)
21
ans =
22
  logical
23
   1
24
>> all(all([x y z]>0))
25
ans =
26
  logical
27
   0
28
% verify if it is a symmetric matrix
29
>> A=[1 2 3
30
2 1 4
31
3 4 1]
32
A =
33
     1     2     3
34
     2     1     4
35
     3     4     1
36
>> all(all(A==A'))
37
ans =
38
  logical
39
   1
40
% verify if it is an upper triangular matrix
41
>> B=[2 3 4
42
0 3 2
43
0 0 1]
44
B =
45
     2     3     4
46
     0     3     2
47
     0     0     1
48
>> any(any(tril(B,-1)))
49
ans =
50
  logical
51
   0
52
>> all(all(B))
53
ans =
54
  logical
55
   0
56
>> all(all(B==triu(B)))
57
ans =
58
  logical
59
   1

Some Others

Functions	Explanations
`isnan(Matrix_name)`	`NaN`: 1 others: 0
`isinf(Matrix_name)`	`inf`: 1 others: 0
`isempty(Matrix_name)`	empty: 1 else: 0
`isequal(Matrix_name1,Matrix_name2)`	equal: 1 else: 0
`isreal(Matrix_name)`	real: 1 else: 0
`isfinite(Matrix_name)`	finite: 1 others: 0

Matrix Creation

Basic Matrix Establishment

1-matrix, 0-matrix and Unit Matrix


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1
% ones
2
>> ones(3)
3
ans =
4
     1     1     1
5
     1     1     1
6
     1     1     1
7
>> ones(4,3,2)
8
ans(:,:,1) =
9
     1     1     1
10
     1     1     1
11
     1     1     1
12
     1     1     1
13
ans(:,:,2) =
14
     1     1     1
15
     1     1     1
16
     1     1     1
17
     1     1     1
18
>> A=[1 2 3 4 5
19
2 3 1 2 5];
20
>> ones(size(A))
21
ans =
22
     1     1     1     1     1
23
     1     1     1     1     1
24
% zeros is similar
25
% eye
26
    % 2-dim only!
27
>> eye(4)
28
ans =
29
     1     0     0     0
30
     0     1     0     0
31
     0     0     1     0
32
     0     0     0     1
33
>> eye(3,4)
34
ans =
35
     1     0     0     0
36
     0     1     0     0
37
     0     0     1     0
38
>> eye(size(A))
39
ans =
40
     1     0     0     0     0
41
     0     1     0     0     0

Random Matrix

rand to generate random numbers between 0 and 1 evenly


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1
>> rand
2
ans =
3
    0.4024
4
>> rand(4)
5
ans =
6
    0.6589    0.1084    0.9620    0.2599
7
    0.9013    0.0361    0.7461    0.9620
8
    0.9954    0.6181    0.6625    0.5402
9
    0.6532    0.5671    0.5233    0.0303
10
>> rand(4,3,2)
11
ans(:,:,1) =
12
    0.6963    0.3302    0.2284
13
    0.5197    0.2297    0.6520
14
    0.0590    0.1139    0.0662
15
    0.8900    0.3109    0.2754
16
ans(:,:,2) =
17
    0.2818    0.6033    0.8486
18
    0.8801    0.7833    0.0506
19
    0.4443    0.1139    0.4662
20
    0.7559    0.9786    0.3257

randn similar to rand, normally distributed random number with unit variance and zero mean