numpy创建单位矩阵和对角矩阵的实例

2023-08-05 06:10:06 71

在学习linearregression时经常处理的数据一般多是矩阵或者n维向量的数据形式，所以必须对矩阵有一定的认识基础。

numpy中创建单位矩阵借助identity（）函数。更为准确的说，此函数创建的是一个n*n的单位数组，返回值的dtype=array数据形式。其中接受的参数有两个，第一个是n值大小，第二个为数据类型，一般为浮点型。单位数组的概念与单位矩阵相同，主对角线元素为1，其他元素均为零，等同于单位1。而要想得到单位矩阵，只要用mat（）函数将数组转换为矩阵即可。

>>>importnumpyasnp
>>>help(np.identity)

Helponfunctionidentityinmodulenumpy:

identity(n,dtype=None)
Returntheidentityarray.

Theidentityarrayisasquarearraywithoneson
themaindiagonal.

Parameters
----------
n:int
Numberofrows(andcolumns)in`n`x`n`output.
dtype:data-type,optional
Data-typeoftheoutput.Defaultsto``float``.

Returns
-------
out:ndarray
`n`x`n`arraywithitsmaindiagonalsettoone,
andallotherelements0.

Examples
--------
>>>np.identity(3)
array([[1.,0.,0.],
[0.,1.,0.],
[0.,0.,1.]])
>>>np.identity(5)

array([[1.,0.,0.,0.,0.],
[0.,1.,0.,0.,0.],
[0.,0.,1.,0.,0.],
[0.,0.,0.,1.,0.],
[0.,0.,0.,0.,1.]])
>>>A=np.mat(np.identity(5))

>>>A

matrix([[1.,0.,0.,0.,0.],
[0.,1.,0.,0.,0.],
[0.,0.,1.,0.,0.],
[0.,0.,0.,1.,0.],
[0.,0.,0.,0.,1.]])

矩阵的运算中还经常使用对角阵，numpy中的对角阵用eye（）函数来创建。eye（）函数接受五个参数，返回一个单位数组。第一个和第二个参数N，M分别对应表示创建数组的行数和列数，当然当你只设定一个值时，就默认了N=M。第三个参数k是对角线指数，跟diagonal中的offset参数是一样的，默认值为0，就是主对角线的方向，上三角方向为正，下三角方向为负，可以取-n到+m的范围。第四个参数是dtype，用于指定元素的数据类型，第五个参数是order，用于排序，有‘C'和‘F'两个参数，默认值为‘C'，为行排序，‘F'为列排序。返回值为一个单位数组。

>>>help(np.eye)

Helponfunctioneyeinmodulenumpy:

eye(N,M=None,k=0,dtype=,order='C')
Returna2-Darraywithonesonthediagonalandzeroselsewhere.

Parameters
----------
N:int
Numberofrowsintheoutput.
M:int,optional
Numberofcolumnsintheoutput.IfNone,defaultsto`N`.
k:int,optional
Indexofthediagonal:0(thedefault)referstothemaindiagonal,
apositivevaluereferstoanupperdiagonal,andanegativevalue
toalowerdiagonal.
dtype:data-type,optional
Data-typeofthereturnedarray.
order:{'C','F'},optional
Whethertheoutputshouldbestoredinrow-major(C-style)or
column-major(Fortran-style)orderinmemory.

..versionadded::1.14.0

Returns
-------
I:ndarrayofshape(N,M)
Anarraywhereallelementsareequaltozero,exceptforthe`k`-th
diagonal,whosevaluesareequaltoone.

SeeAlso
--------
identity:(almost)equivalentfunction
diag:diagonal2-Darrayfroma1-Darrayspecifiedbytheuser.

Examples
--------
>>>np.eye(2,dtype=int)
array([[1,0],
[0,1]])
>>>np.eye(3,k=1)
array([[0.,1.,0.],
[0.,0.,1.],
[0.,0.,0.]])

numpy中的diagonal（）方法可以对n*n的数组和方阵取对角线上的元素，diagonal（）接受三个参数。第一个offset参数是主对角线的方向，默认值为0是主对角线，上三角方向为正，下三角方向为负，可以取-n到+n的范围。第二个参数和第三个参数是在数组大于2维时指定一个2维数组时使用，默认值axis1=0，axis2=1。

>>>help(A.diagonal)

Helponbuilt-infunctiondiagonal:

diagonal(...)methodofnumpy.matrixinstance
a.diagonal(offset=0,axis1=0,axis2=1)

Returnspecifieddiagonals.InNumPy1.9thereturnedarrayisa
read-onlyviewinsteadofacopyasinpreviousNumPyversions.In
afutureversiontheread-onlyrestrictionwillberemoved.

Referto:func:`numpy.diagonal`forfulldocumentation.

SeeAlso
--------
numpy.diagonal:equivalentfunction
>>>help(np.diagonal)

Helponfunctiondiagonalinmodulenumpy:

diagonal(a,offset=0,axis1=0,axis2=1)
Returnspecifieddiagonals.

If`a`is2-D,returnsthediagonalof`a`withthegivenoffset,
i.e.,thecollectionofelementsoftheform``a[i,i+offset]``.If
`a`hasmorethantwodimensions,thentheaxesspecifiedby`axis1`
and`axis2`areusedtodeterminethe2-Dsub-arraywhosediagonalis
returned.Theshapeoftheresultingarraycanbedeterminedby
removing`axis1`and`axis2`andappendinganindextotherightequal
tothesizeoftheresultingdiagonals.

InversionsofNumPypriorto1.7,thisfunctionalwaysreturnedanew,
independentarraycontainingacopyofthevaluesinthediagonal.

InNumPy1.7and1.8,itcontinuestoreturnacopyofthediagonal,
butdependingonthisfactisdeprecated.Writingtotheresulting
arraycontinuestoworkasitusedto,butaFutureWarningisissued.

StartinginNumPy1.9itreturnsaread-onlyviewontheoriginalarray.
Attemptingtowritetotheresultingarraywillproduceanerror.

Insomefuturerelease,itwillreturnaread/writeviewandwritingto
thereturnedarraywillalteryouroriginalarray.Thereturnedarray
willhavethesametypeastheinputarray.

Ifyoudon'twritetothearrayreturnedbythisfunction,thenyoucan
justignorealloftheabove.

Ifyoudependonthecurrentbehavior,thenwesuggestcopyingthe
returnedarrayexplicitly,i.e.,use``np.diagonal(a).copy()``instead
ofjust``np.diagonal(a)``.Thiswillworkwithbothpastandfuture
versionsofNumPy.

Parameters
----------
a:array_like
Arrayfromwhichthediagonalsaretaken.
offset:int,optional
Offsetofthediagonalfromthemaindiagonal.Canbepositiveor
negative.Defaultstomaindiagonal(0).
axis1:int,optional
Axistobeusedasthefirstaxisofthe2-Dsub-arraysfromwhich
thediagonalsshouldbetaken.Defaultstofirstaxis(0).
axis2:int,optional
Axistobeusedasthesecondaxisofthe2-Dsub-arraysfrom
whichthediagonalsshouldbetaken.Defaultstosecondaxis(1).

Returns
-------
array_of_diagonals:ndarray
If`a`is2-D,thena1-Darraycontainingthediagonalandofthe
sametypeas`a`isreturnedunless`a`isa`matrix`,inwhichcase
a1-Darrayratherthana(2-D)`matrix`isreturnedinorderto
maintainbackwardcompatibility.

If``a.ndim>2``,thenthedimensionsspecifiedby`axis1`and`axis2`
areremoved,andanewaxisinsertedattheendcorrespondingtothe
diagonal.

Raises
------
ValueError
Ifthedimensionof`a`islessthan2.

SeeAlso
--------
diag:MATLABwork-a-likefor1-Dand2-Darrays.
diagflat:Creatediagonalarrays.
trace:Sumalongdiagonals.

Examples
--------
>>>a=np.arange(4).reshape(2,2)
>>>a
array([[0,1],
[2,3]])
>>>a.diagonal()
array([0,3])
>>>a.diagonal(1)
array([1])

A3-Dexample:

>>>a=np.arange(8).reshape(2,2,2);a
array([[[0,1],
[2,3]],
[[4,5],
[6,7]]])
>>>a.diagonal(0,#Maindiagonalsoftwoarrayscreatedbyskipping
...0,#acrosstheouter(left)-mostaxislastand
...1)#the"middle"(row)axisfirst.
array([[0,6],
[1,7]])

Thesub-arrayswhosemaindiagonalswejustobtained;notethateach
correspondstofixingtheright-most(column)axis,andthatthe
diagonalsare"packed"inrows.

>>>a[:,:,0]#maindiagonalis[06]
array([[0,2],
[4,6]])
>>>a[:,:,1]#maindiagonalis[17]
array([[1,3],
[5,7]])
>>>A=np.random.randint(low=5,high=30,size=(5,5))

>>>A

array([[25,15,26,6,22],
[27,14,22,16,21],
[22,17,10,14,25],
[11,9,27,20,6],
[24,19,19,26,14]])
>>>A.diagonal()

array([25,14,10,20,14])
>>>A.diagonal(offset=1)

array([15,22,14,6])
>>>A.diagonal(offset=-2)

array([22,9,19])

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numpy创建单位矩阵和对角矩阵的实例

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