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PositiveSemidefiniteMatrixQ
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PositiveSemidefiniteMatrixQ

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如果 m 是显式半正定矩阵，给出 True，否则给出 False.

更多信息和选项

  • 如果对于所有向量 x，Re[Conjugate[x].m.x]≥0 成立，则矩阵 m 是半正定的. »
  • PositiveSemidefiniteMatrixQ 适用于符号以及数值矩阵.
  • 对于近似矩阵，选项 Tolerance->t 可用于表明将所有满足 λ≤t λmax 的特征值 λ 视为零，其中 λmax 是幅值最大的特征值.
  • 选项 Tolerance 具有默认值 Automatic.

范例

打开所有单元关闭所有单元

基本范例  (2)常见实例总结

测试 2×2 的实矩阵是否为显式半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-k5tb4m
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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1iqf8o
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Out[2]=2

这意味着对于所有向量 ，二次型 成立：

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In[3]:=3
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-gntl81
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Out[3]=3

可视化二次型的值：

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In[4]:=4
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0bvue1
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Out[4]=4

测试 3×3 Hermitian 矩阵是否为半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-znfdke
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Out[1]=1

范围  (10)标准用法实例范围调查

基本用法  (6)

测试一个实机器精度矩阵是否为显式半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1kq4tu
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Out[1]=1

测试一个复矩阵否为半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-5fazgj
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Out[1]=1

测试一个精确矩阵否为半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-tfdinn
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Out[1]=1

用 PositiveSemidefiniteMatrixQ 测试任意精度的矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-eoeqf9
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Out[1]=1

随机矩阵通常不是半正定矩阵：

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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ow584e
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Out[2]=2

用 PositiveSemidefiniteMatrixQ 测试符号矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-62ooyj
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Out[1]=1

b=-TemplateBox[{a}, Conjugate] 时矩阵为半正定矩阵：

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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-rni58c
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Out[2]=2

PositiveSemidefiniteMatrixQ 可高效处理大型数值矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1gk6wb
Direct link to example
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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-c5kyvx
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-nb2gv9
Direct link to example
Out[3]=3

特殊矩阵  (4)

用 PositiveSemidefiniteMatrixQ 测试稀疏矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0lcs36
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-cvjhfs
Direct link to example
Out[2]=2

用 PositiveSemidefiniteMatrixQ 测试结构矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-my6pw
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-n6jq9t
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Out[2]=2

单位矩阵是半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-tpv6mn
Direct link to example
Out[1]=1

HilbertMatrix 是半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-slp5k9
Direct link to example
Out[1]=1

选项  (1)各选项的常用值和功能

Tolerance  (1)

产生具有阶数为 10-12 的随机扰动的实数对角矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-35lu76
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-mrsvrl
Direct link to example
Out[2]=2

调整选项 Tolerance 接受矩阵作为半正定矩阵：

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In[3]:=3
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-cbjh4n
Direct link to example
Out[3]=3

应用  (13)用该函数可以解决的问题范例

半正定矩阵的几何与代数性质  (5)

考虑一个 2×2 的实半正定矩阵及其相关的实二次型 q=TemplateBox[{x}, Transpose].m.x：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-xq3amu
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-7eaz20
Direct link to example

因为 是正定矩阵，所以水平集是椭圆：

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In[4]:=4
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-2bl8ys
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Out[4]=4

的图则是向上的椭圆抛物面：

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In[5]:=5
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-mw6kb
Direct link to example
Out[5]=5

但是，椭圆可以退化，变成线：

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In[5]:=5
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-z3oc9l
Direct link to example
Out[6]=6
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In[7]:=7
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-77xx63
Direct link to example
Out[7]=7

的图则变为抛物线上的圆柱体：i

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In[9]:=9
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-589e5j
Direct link to example
Out[9]=9

在更极端的情况下，由于 ，水平集可以是整个平面：

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In[8]:=8
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-upvrmb
Direct link to example
Out[9]=9
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In[10]:=10
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-e6yjz3
Direct link to example
Out[10]=10

对于 的实半正定矩阵，水平集是 -椭圆体：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-lrsik1
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-xutlb2
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-zxn0wi
Direct link to example
Out[3]=3

在三维空间中，可以退化为椭圆上的圆柱体：

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In[4]:=4
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-i4nhjp
Direct link to example
Out[4]=4
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In[5]:=5
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-4fnyml
Direct link to example
Out[5]=5

以及平面：

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In[6]:=6
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1b2grz
Direct link to example
Out[6]=6
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In[7]:=7
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-x0igid
Direct link to example
Out[7]=7

通过 q=TemplateBox[{x}, ConjugateTranspose].m.x，Hermitian 矩阵定义了一个实二次型：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-t4honb
Direct link to example
Out[1]=1

如果 是半正定矩阵，对于所有输入， 非负：

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In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-8p80us
Direct link to example
Out[2]=2

输入为实数的情况下可视化 ：

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In[3]:=3
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-wvuonx
Direct link to example
Out[3]=3

对于实值矩阵 ，只有对称的部分决定 是否为半正定矩阵. 写为 ， 为对称部分， 为非对称部分：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-343mgv
Direct link to example
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-i3xo0f
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-w3jv4l
Direct link to example
Out[3]=3

由于 为实对称矩阵，TemplateBox[{{(, {TemplateBox[{x}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], ., s, ., x}, )}}, Conjugate]=TemplateBox[{{(, {TemplateBox[{x}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], ., s, ., x}, )}}, ConjugateTranspose]=TemplateBox[{x}, ConjugateTranspose].TemplateBox[{s}, ConjugateTranspose].TemplateBox[{{(, TemplateBox[{x}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], )}}, ConjugateTranspose]=TemplateBox[{x}, ConjugateTranspose].s.x，意味着 TemplateBox[{x}, ConjugateTranspose].s.x 是实矩阵：

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In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-jf2jdt
Direct link to example
Out[4]=4

同样，由于 为实非对称矩阵，有 TemplateBox[{{(, {TemplateBox[{x}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], ., a, ., x}, )}}, Conjugate]=-TemplateBox[{x}, ConjugateTranspose].a.x 或 TemplateBox[{x}, ConjugateTranspose].a.x 为虚矩阵：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ri3cbu
Direct link to example
Out[5]=5

因此，Re(TemplateBox[{x}, ConjugateTranspose].m.x)=TemplateBox[{x}, ConjugateTranspose].s.x，当且仅当 为半正定矩阵， 才是半正定矩阵：

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In[6]:=6
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-cg0348
Direct link to example
Out[6]=6

对于复矩阵 ，只有 Hermitian 部分决定 是否为半正定矩阵. 写为 ， 是 Hermitian 矩阵， 为反埃尔米特矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-7in8hm
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-tq6566
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1i72sc
Direct link to example
Out[3]=3

由于 是 Hermitian 矩阵，TemplateBox[{{(, {TemplateBox[{x}, ConjugateTranspose], ., h, ., x}, )}}, Conjugate]=TemplateBox[{{(, {TemplateBox[{x}, ConjugateTranspose], ., h, ., x}, )}}, ConjugateTranspose]=TemplateBox[{x}, ConjugateTranspose].TemplateBox[{h}, ConjugateTranspose].TemplateBox[{{(, TemplateBox[{x}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], )}}, ConjugateTranspose]=TemplateBox[{x}, ConjugateTranspose].h.x，意味着 TemplateBox[{x}, ConjugateTranspose].h.x 是实矩阵：

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In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-33o78l
Direct link to example
Out[4]=4

同样，由于 为反埃尔米特矩阵，有 TemplateBox[{{(, {TemplateBox[{x}, ConjugateTranspose, SyntaxForm -> SuperscriptBox], ., a, ., x}, )}}, Conjugate]=-TemplateBox[{x}, ConjugateTranspose].a.x 或 TemplateBox[{x}, ConjugateTranspose].a.x 是虚矩阵：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ww36rj
Direct link to example
Out[5]=5

因此，Re(TemplateBox[{x}, ConjugateTranspose].m.x)=TemplateBox[{x}, ConjugateTranspose].h.x，当且仅当 为半正定矩阵， 才是半正定矩阵：

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In[6]:=6
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-7zx7p2
Direct link to example
Out[6]=6

半正定矩阵的来源  (8)

实 Covariance 矩阵总是对称的和半正定的：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-r24rp8
Direct link to example
Out[1]=1

复矩阵是 Hermitian 矩阵：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-z9e7k3
Direct link to example
Out[2]=2

为方阵，计算 SingularValueDecomposition[m]：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ugfzlm
Direct link to example

矩阵式半正定矩阵：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-pu6p09
Direct link to example
Out[2]=2

Gram 矩阵是 个向量的点积组成的 对称矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-q5c30y
Direct link to example

Gram 矩阵一定是半正定矩阵：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ernk9
Direct link to example
Out[2]=2

从 WishartMatrixDistribution 得到的矩阵是实对称半正定矩阵：

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In[1]:=1
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https://wolfram.com/xid/0bhf3wgupbucv1cods4z-gf7yyg
Direct link to example
Out[1]=1

Hermitian 矩阵的平方是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0bx5p8
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-4r8vsp
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-coq2u3
Direct link to example
Out[3]=3

每个反厄米特矩阵都是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-d9orwc
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-co29v5
Direct link to example
Out[2]=2

Lehmer 矩阵是对称半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-11h6uz
Direct link to example
Out[1]=1

它的逆矩阵是三对角矩阵，同时也是对称半正定矩阵：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-dr1ppf
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-hpm29l
Direct link to example
Out[3]=3

矩阵 Min[i,j] 一定是对称半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-r3v5d2
Direct link to example
Out[1]=1

它的逆矩阵是三对角矩阵，同时也是对称半正定矩阵：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-uj576h
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-wpbkgs
Direct link to example
Out[3]=3

属性和关系  (13)函数的属性及与其他函数的关联

对于任何不是矩阵的 x，PositiveSemidefiniteMatrixQ[x] 返回 False：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-52egbw
Direct link to example
Out[1]=1

如果对于所有向量 ，Re(TemplateBox[{x}, ConjugateTranspose].m.x)>=0，则矩阵 是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-uc2nl4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-4rtpxr
Direct link to example
Out[2]=2

与 Im(TemplateBox[{x}, ConjugateTranspose].m.x) 的正负无关：

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In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0aaw2u
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-5s7mky
Direct link to example
Out[4]=4

当且仅当实矩阵的对称部分是半正定的，实矩阵 才是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-gur7yw
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-32pn0w
Direct link to example
Out[2]=2

一般情况下，当且仅当矩阵的 Hermitian 部分是半正定的，矩阵 才是半正定矩阵：

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In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1woht6
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-v0nw29
Direct link to example
Out[4]=4

当且仅当实对称矩阵的特征值都是非负的，实对称矩阵才是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0gbqgh
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-odzxnu
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-mi6tly
Direct link to example
Out[3]=3

该陈述更普遍适用于 Hermitian 矩阵：

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In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-gegyod
Direct link to example
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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-olr87h
Direct link to example
Out[5]=5

普通矩阵的所有特征值可以非负而不是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-2wlag4
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-pnkm16
Direct link to example
Out[2]=2

同样，一个矩阵可以是半正定的，但没有非负特征值：

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In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-i7vtvh
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-pordaa
Direct link to example
Out[4]=4

失败是特征值为复数造成的：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ymsd52
Direct link to example
Out[5]=5

半正定矩阵的特征值的实部必须非负：

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In[6]:=6
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-wktjrm
Direct link to example
Out[6]=6

当且仅当对角元素有非负实部，对角矩阵才是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-6i3wqm
Direct link to example
Out[1]=1
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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-5cykp8
Direct link to example
Out[2]=2

半正定矩阵的通用形式为 u.d.TemplateBox[{u}, ConjugateTranspose]+a，其中， 为对角半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-5mtsrr
Direct link to example
Out[1]=1

将 划分为 Hermitian 和反厄米特部分：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1lhk03
Direct link to example
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In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ros799
Direct link to example

根据谱定理，可用 JordanDecomposition 将 酉对角化：

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In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-5xk4eo
Direct link to example

矩阵 是对角矩阵，且对角元素非负：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-e7l3ru
Direct link to example

矩阵 是酉矩阵：

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In[6]:=6
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-uu247o
Direct link to example
Out[6]=6

验证 m=u.d.TemplateBox[{u}, ConjugateTranspose]+a：

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In[7]:=7
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-s4ntwc
Direct link to example
Out[7]=7

当且仅当 是半负定矩阵，矩阵 才是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-6c25gj
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-hrrwad
Direct link to example
Out[2]=2

正定矩阵一定是半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-umljb8
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-qr2db6
Direct link to example
Out[2]=2

有非正定的半正定矩阵：

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In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-umgbzb
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-ubfv0t
Direct link to example
Out[4]=4

半正定矩阵不可能是不定或半负定矩阵：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-svo4k1
Direct link to example
Out[5]=5

实对称半正定矩阵的行列式和迹非负：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-wm94q3
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-6g4zxu
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-unnzjp
Direct link to example
Out[3]=3

这同样适用于半正定 Hermitian 矩阵：

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In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-1mwrnq
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-gn4gq1
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-6qsx6k
Direct link to example
Out[6]=6

实对称半正定矩阵 有唯一定义的平方根 ，使得 成立：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-h57z2p
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-pjy0vj
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-cicqaf
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-f456sv
Direct link to example
Out[4]=4

平方根 是半正定实对称矩阵：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-2pmo1x
Direct link to example
Out[5]=5

半正定厄米特矩阵 有唯一定义的平方根 ，使得 成立：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0zjggt
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-8z4qk6
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-7bxtuv
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-xgm4wo
Direct link to example
Out[4]=4

平方根 是半正定厄米特矩阵：

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In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-d8y78b
Direct link to example
Out[5]=5

两个对称半正定矩阵的 Kronecker 积是对称半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-luvzz6
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-nhifmq
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-78dgkz
Direct link to example
Out[3]=3

用一个负定矩阵替换其中的一个矩阵，将得到一个半负定矩阵：

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In[4]:=4
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-95pre
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-bi9dt6
Direct link to example
Out[5]=5

可能存在的问题  (2)常见隐患和异常行为

CholeskyDecomposition 不用于对称或者奇异的 Hermitian 半正定矩阵：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-eap5hh
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-fh02j4
Direct link to example
Out[3]=3

PositiveSemidefiniteMatrixQ 将给出 False，除非它可以证明一个符号矩阵是半正定的：

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In[1]:=1
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-mht1w7
Direct link to example
Out[1]=1

同时使用 Eigenvalues 和 Reduce 可给出更精确的结果：

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In[2]:=2
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-0dkk5z
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0bhf3wgupbucv1cods4z-jri0ni
Direct link to example
Out[3]=3
Wolfram Research (2014)，PositiveSemidefiniteMatrixQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html.
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✖
Wolfram Research (2014)，PositiveSemidefiniteMatrixQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html.

文本

Wolfram Research (2014)，PositiveSemidefiniteMatrixQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html.

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✖
Wolfram Research (2014)，PositiveSemidefiniteMatrixQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html.

CMS

Wolfram 语言. 2014. "PositiveSemidefiniteMatrixQ." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html.

Copy to clipboard.
✖
Wolfram 语言. 2014. "PositiveSemidefiniteMatrixQ." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html.

APA

Wolfram 语言. (2014). PositiveSemidefiniteMatrixQ. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html 年

Copy to clipboard.
✖
Wolfram 语言. (2014). PositiveSemidefiniteMatrixQ. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html 年

BibTeX

@misc{reference.wolfram_2025_positivesemidefinitematrixq, author="Wolfram Research", title="{PositiveSemidefiniteMatrixQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html}", note=[Accessed: 30-March-2025 ]}

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✖
@misc{reference.wolfram_2025_positivesemidefinitematrixq, author="Wolfram Research", title="{PositiveSemidefiniteMatrixQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html}", note=[Accessed: 30-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_positivesemidefinitematrixq, organization={Wolfram Research}, title={PositiveSemidefiniteMatrixQ}, year={2014}, url={https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html}, note=[Accessed: 30-March-2025 ]}

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✖
@online{reference.wolfram_2025_positivesemidefinitematrixq, organization={Wolfram Research}, title={PositiveSemidefiniteMatrixQ}, year={2014}, url={https://reference.wolfram.com/language/ref/PositiveSemidefiniteMatrixQ.html}, note=[Accessed: 30-March-2025 ]}