The matrix of ones is in the mathematics a matrix whose elements are all equal to the number one (or the identity of the underlying ring are). A ones matrix that consists of only one row or column is also called a one vector . Every one matrix can be represented as a dyadic product of one vectors. In the matrix ring with the matrix addition and the Hadamard product , the one matrix is the neutral element . Important key figures and powers of single matrices can be calculated explicitly. The one matrix and the one vector must not be confused with the unit matrix and the unit vector .
definition
If a ring is one element , then the one matrix is defined as
R.
{\ displaystyle R}
1
{\ displaystyle 1}
1
1
m
n
∈
R.
m
×
n
{\ displaystyle 1 \! \! 1_ {mn} \ in R ^ {m \ times n}}
1
1
m
n
=
(
1
⋯
1
⋮
⋱
⋮
1
⋯
1
)
{\ displaystyle 1 \! \! 1_ {mn} = {\ begin {pmatrix} 1 & \ cdots & 1 \\\ vdots & \ ddots & \ vdots \\ 1 & \ cdots & 1 \ end {pmatrix}}}
.
A one matrix consisting of only one row or column is also called a one vector and is designated with . If the dimension of the one matrix becomes clear from the context and there is no possibility of confusion, the indices are also omitted and only written. Based on standard matrices , which are often referred to as, single matrices are also noted by.
1
1
n
{\ displaystyle 1 \! \! 1_ {n}}
1
1
{\ displaystyle 1 \! \! 1}
I.
{\ displaystyle I}
J
{\ displaystyle J}
Examples
If the field of real numbers is one and denotes the number one , then examples of one vectors and matrices are:
R.
{\ displaystyle R}
1
{\ displaystyle 1}
1
1
2
=
(
1
1
)
,
1
1
3
=
(
1
1
1
)
,
1
1
22nd
=
(
1
1
1
1
)
,
1
1
33
=
(
1
1
1
1
1
1
1
1
1
)
,
1
1
24
=
(
1
1
1
1
1
1
1
1
)
{\ displaystyle 1 \! \! 1_ {2} = {\ begin {pmatrix} 1 \\ 1 \ end {pmatrix}}, 1 \! \! 1_ {3} = {\ begin {pmatrix} 1 \\ 1 \\ 1 \ end {pmatrix}}, 1 \! \! 1_ {22} = {\ begin {pmatrix} 1 & 1 \\ 1 & 1 \ end {pmatrix}}, 1 \! \! 1_ {33} = {\ begin {pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \ end {pmatrix}}, 1 \! \! 1_ {24} = {\ begin {pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \ end {pmatrix}}}
Let the null ring be , then the following matrices are also examples of single matrices:
R.
{\ displaystyle R}
1
1
2
=
(
0
1
)
,
1
1
3
=
(
1
0
0
)
,
1
1
22nd
=
(
1
1
0
0
)
,
1
1
33
=
(
1
1
1
1
0
1
1
1
0
)
,
1
1
24
=
(
1
1
1
0
0
1
0
1
)
{\ displaystyle 1 \! \! 1_ {2} = {\ begin {pmatrix} 0 \\ 1 \ end {pmatrix}}, 1 \! \! 1_ {3} = {\ begin {pmatrix} 1 \\ 0 \\ 0 \ end {pmatrix}}, 1 \! \! 1_ {22} = {\ begin {pmatrix} 1 & 1 \\ 0 & 0 \ end {pmatrix}}, 1 \! \! 1_ {33} = {\ begin {pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \ end {pmatrix}}, 1 \! \! 1_ {24} = {\ begin {pmatrix} 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \ end {pmatrix}}}
Note: The terms zero matrix and one matrix coincide in the zero ring . In fact, even every matrix above the zero ring is a one matrix (and a zero matrix).
properties
Algebraic properties
A ones matrix can also be represented as a dyadic product of one vectors:
1
1
m
n
=
1
1
m
⊗
1
1
n
=
1
1
m
⋅
(
1
1
n
)
T
{\ displaystyle 1 \! \! 1_ {mn} = 1 \! \! 1_ {m} \ otimes 1 \! \! 1_ {n} = 1 \! \! 1_ {m} \ cdot (1 \! \ ! 1_ {n}) ^ {T}}
.
The transpose of a one matrix is again a one matrix, so
(
1
1
m
n
)
T
=
1
1
n
m
{\ displaystyle (1 \! \! 1_ {mn}) ^ {T} = 1 \! \! 1_ {nm}}
.
The one matrix is also the neutral element in the matrix ring , where the matrix addition and the Hadamard product . This applies to all matrices
1
1
m
n
{\ displaystyle 1 \! \! 1_ {mn}}
(
R.
m
×
n
,
+
,
∘
)
{\ displaystyle (R ^ {m \ times n}, +, \ circ)}
A.
+
B.
{\ displaystyle A + B}
A.
∘
B.
{\ displaystyle A \ circ B}
A.
∈
R.
m
×
n
{\ displaystyle A \ in R ^ {m \ times n}}
A.
∘
1
1
m
n
=
1
1
m
n
∘
A.
=
A.
{\ displaystyle A \ circ 1 \! \! 1_ {mn} = 1 \! \! 1_ {mn} \ circ A = A}
.
Rank, determinant, trace
If now is a field , then the rank is a one matrix
R.
{\ displaystyle R}
rank
(
1
1
m
n
)
=
1
{\ displaystyle \ operatorname {rank} (1 \! \! 1_ {mn}) = 1}
.
The determinant of a square one matrix is then
det
(
1
1
n
n
)
=
{
0
if
n
>
1
,
1
if
n
=
1.
{\ displaystyle \ det (1 \! \! 1_ {nn}) = {\ begin {cases} 0 & {\ text {falls}} ~ n> 1, \\ 1 & {\ text {falls}} ~ n = 1 . \ end {cases}}}
The trace of a square ones matrix over the real or complex numbers is
track
(
1
1
n
n
)
=
n
{\ displaystyle \ operatorname {spur} (1 \! \! 1_ {nn}) = n}
.
Eigenvalues
The characteristic polynomial of a real or complex one matrix results as
1
n
n
{\ displaystyle 1_ {nn}}
χ
(
λ
)
=
λ
n
-
1
(
λ
-
n
)
{\ displaystyle \ chi (\ lambda) = \ lambda ^ {n-1} (\ lambda -n)}
.
The eigenvalues are accordingly
λ
1
=
n
{\ displaystyle \ lambda _ {1} = n}
and .
λ
2
=
...
=
λ
n
=
0
{\ displaystyle \ lambda _ {2} = \ ldots = \ lambda _ {n} = 0}
Associated eigenvectors are
(
1
,
...
,
1
)
T
{\ displaystyle (1, \ ldots, 1) ^ {T}}
and .
(
1
,
-
1
,
0
,
...
,
0
)
T
,
...
,
(
0
,
...
,
0
,
1
,
-
1
)
T
{\ displaystyle (1, -1,0, \ ldots, 0) ^ {T}, \ ldots, (0, \ ldots, 0,1, -1) ^ {T}}
Products
The following applies to the product of two real or complex single matrices of suitable size
1
1
m
n
⋅
1
1
n
O
=
n
⋅
1
1
m
O
{\ displaystyle 1 \! \! 1_ {mn} \ cdot 1 \! \! 1_ {no} = n \ cdot 1 \! \! 1_ {mo}}
.
This calculates the -th power of a square one matrix for as
k
{\ displaystyle k}
k
≥
1
{\ displaystyle k \ geq 1}
(
1
1
n
n
)
k
=
n
k
-
1
1
1
n
n
{\ displaystyle (1 \! \! 1_ {nn}) ^ {k} = n ^ {k-1} 1 \! \! 1_ {nn}}
.
Hence the matrix is idempotent , that is
1
n
1
1
n
n
{\ displaystyle {\ tfrac {1} {n}} 1 \! \! 1_ {nn}}
1
n
1
1
n
n
⋅
1
n
1
1
n
n
=
1
n
1
1
n
n
{\ displaystyle {\ tfrac {1} {n}} 1 \! \! 1_ {nn} \ cdot {\ tfrac {1} {n}} 1 \! \! 1_ {nn} = {\ tfrac {1} {n}} 1 \! \! 1_ {nn}}
.
The following applies for the matrix exponential of the one matrix
exp
(
1
1
n
n
)
=
∑
k
=
0
∞
(
1
1
n
n
)
k
k
!
=
I.
n
+
∑
k
=
1
∞
n
k
-
1
k
!
⋅
1
1
n
n
=
I.
n
+
e
n
-
1
n
⋅
1
1
n
n
{\ displaystyle \ exp (1 \! \! 1_ {nn}) = \ sum _ {k = 0} ^ {\ infty} {\ frac {(1 \! \! 1_ {nn}) ^ {k}} {k!}} = I_ {n} + \ sum _ {k = 1} ^ {\ infty} {\ frac {n ^ {k-1}} {k!}} \ cdot 1 \! \! 1_ { nn} = I_ {n} + {\ frac {e ^ {n} -1} {n}} \ cdot 1 \! \! 1_ {nn}}
,
wherein the unit matrix of the size and the Euler number are.
I.
n
{\ displaystyle I_ {n}}
n
{\ displaystyle n}
e
{\ displaystyle e}
programming
In the numerical software package MATLAB , the one matrix is ones(m,n)
generated by the function .
literature
Karsten Schmidt, Götz Trenkler: Introduction to Modern Matrix Algebra . Springer, 2006, ISBN 3-540-33008-9 .
Individual evidence
↑ Schmidt, Trenkler: Introduction to Modern Matrix Algebra . S. 27-28 .
↑ Christoph W. Überhuber, Stefan Katzenbeisser, Dirk Praetorius: MATLAB 7: An Introduction . Springer, 2007, p. 18 .
Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">