Applications Moore–Penrose inverse
1 applications
1.1 linear least-squares
1.2 obtaining solutions of linear system
1.3 minimum norm solution linear system
1.4 condition number
applications
linear least-squares
the pseudoinverse provides least squares solution system of linear equations.
a
∈
m
(
m
,
n
;
k
)
{\displaystyle a\in \mathrm {m} (m,n;k)\,\!}
, given system of linear equations
a
x
=
b
,
{\displaystyle ax=b,\,}
in general, vector
x
{\displaystyle x}
solves system may not exist, or if 1 exist, may not unique. pseudoinverse solves least-squares problem follows:
∀
x
∈
k
n
{\displaystyle \forall x\in k^{n}\,\!}
, have
∥
a
x
−
b
∥
2
≥
∥
a
z
−
b
∥
2
{\displaystyle \|ax-b\|_{2}\geq \|az-b\|_{2}}
z
=
a
+
b
{\displaystyle z=a^{+}b}
,
∥
⋅
∥
2
{\displaystyle \|\cdot \|_{2}}
denotes euclidean norm. weak inequality holds equality if , if
x
=
a
+
b
+
(
i
−
a
+
a
)
w
{\displaystyle x=a^{+}b+(i-a^{+}a)w}
vector w; provides infinitude of minimizing solutions unless has full column rank, in case
(
i
−
a
+
a
)
{\displaystyle (i-a^{+}a)}
0 matrix. solution minimum euclidean norm
z
.
{\displaystyle z.}
this result extended systems multiple right-hand sides, when euclidean norm replaced frobenius norm. let
b
∈
m
(
m
,
p
;
k
)
{\displaystyle b\in \mathrm {m} (m,p;k)}
.
∀
x
∈
m
(
n
,
p
;
k
)
{\displaystyle \forall x\in \mathrm {m} (n,p;k)\,\!}
, have
∥
a
x
−
b
∥
f
≥
∥
a
z
−
b
∥
f
{\displaystyle \|ax-b\|_{\mathrm {f} }\geq \|az-b\|_{\mathrm {f} }}
z
=
a
+
b
{\displaystyle z=a^{+}b}
,
∥
⋅
∥
f
{\displaystyle \|\cdot \|_{\mathrm {f} }}
denotes frobenius norm.
obtaining solutions of linear system
if linear system
a
x
=
b
{\displaystyle ax=b\,}
has solutions, given by
x
=
a
+
b
+
[
i
−
a
+
a
]
w
{\displaystyle x=a^{+}b+[i-a^{+}a]w}
for arbitrary vector w. solution(s) exist if , if
a
a
+
b
=
b
{\displaystyle aa^{+}b=b}
. if latter holds, solution unique if , if has full column rank, in case
[
i
−
a
+
a
]
{\displaystyle [i-a^{+}a]}
0 matrix. if solutions exist not have full column rank, have indeterminate system, of infinitude of solutions given last equation. solution connected udwadia–kalaba equation of classical mechanics forces of constraint not obey d alembert s principle.
minimum norm solution linear system
for linear systems
a
x
=
b
,
{\displaystyle ax=b,\,}
non-unique solutions (such under-determined systems), pseudoinverse may used construct solution of minimum euclidean norm
∥
x
∥
2
{\displaystyle \|x\|_{2}}
among solutions.
if
a
x
=
b
{\displaystyle ax=b\,}
satisfiable, vector
z
=
a
+
b
{\displaystyle z=a^{+}b}
solution, , satisfies
∥
z
∥
2
≤
∥
x
∥
2
{\displaystyle \|z\|_{2}\leq \|x\|_{2}}
solutions.
this result extended systems multiple right-hand sides, when euclidean norm replaced frobenius norm. let
b
∈
m
(
m
,
p
;
k
)
{\displaystyle b\in \mathrm {m} (m,p;k)\,\!}
.
if
a
x
=
b
{\displaystyle ax=b\,}
satisfiable, matrix
z
=
a
+
b
{\displaystyle z=a^{+}b}
solution, , satisfies
∥
z
∥
f
≤
∥
x
∥
f
{\displaystyle \|z\|_{\mathrm {f} }\leq \|x\|_{\mathrm {f} }}
solutions.
condition number
using pseudoinverse , matrix norm, 1 can define condition number matrix:
cond
(
a
)
=
∥
a
∥
∥
a
+
∥
.
{\displaystyle {\mbox{cond}}(a)=\|a\|\|a^{+}\|.\ }
a large condition number implies problem of finding least-squares solutions corresponding system of linear equations ill-conditioned in sense small errors in entries of can lead huge errors in entries of solution.
Comments
Post a Comment