Special cases Moore–Penrose inverse
1 special cases
1.1 scalars
1.2 vectors
1.3 linearly independent columns
1.4 linearly independent rows
1.5 orthonormal columns or rows
1.6 orthogonal projection matrices
1.7 circulant matrices
special cases
scalars
it possible define pseudoinverse scalars , vectors. amounts treating these matrices. pseudoinverse of scalar x 0 if x 0 , reciprocal of x otherwise:
x
+
=
{
0
,
if
x
=
0
;
x
−
1
,
otherwise
.
{\displaystyle x^{+}=\left\{{\begin{matrix}0,&{\mbox{if }}x=0;\\x^{-1},&{\mbox{otherwise}}.\end{matrix}}\right.}
vectors
the pseudoinverse of null (all zero) vector transposed null vector. pseudoinverse of non-null vector conjugate transposed vector divided squared magnitude:
x
+
=
{
0
t
,
if
x
=
0
;
x
∗
x
∗
x
,
otherwise
.
{\displaystyle x^{+}=\left\{{\begin{matrix}0^{\mathrm {t} },&{\mbox{if }}x=0;\\{x^{*} \over x^{*}x},&{\mbox{otherwise}}.\end{matrix}}\right.}
linearly independent columns
if columns of
a
{\displaystyle a\,\!}
linearly independent (so
m
≥
n
{\displaystyle m\geq n}
),
a
∗
a
{\displaystyle a^{*}a\,\!}
invertible. in case, explicit formula is:
a
+
=
(
a
∗
a
)
−
1
a
∗
{\displaystyle a^{+}=(a^{*}a)^{-1}a^{*}\,\!}
.
it follows
a
+
{\displaystyle a^{+}\,\!}
left inverse of
a
{\displaystyle a\,\!}
:
a
+
a
=
i
n
{\displaystyle a^{+}a=i_{n}\,\!}
.
linearly independent rows
if rows of
a
{\displaystyle a\,\!}
linearly independent (so
m
≤
n
{\displaystyle m\leq n}
),
a
a
∗
{\displaystyle aa^{*}}
invertible. in case, explicit formula is:
a
+
=
a
∗
(
a
a
∗
)
−
1
{\displaystyle a^{+}=a^{*}(aa^{*})^{-1}\,\!}
.
it follows
a
+
{\displaystyle a^{+}\,\!}
right inverse of
a
{\displaystyle a\,\!}
:
a
a
+
=
i
m
{\displaystyle aa^{+}=i_{m}\,\!}
.
orthonormal columns or rows
this special case of either full column rank or full row rank (treated above). if
a
{\displaystyle a\,\!}
has orthonormal columns (
a
∗
a
=
i
n
{\displaystyle a^{*}a=i_{n}\,\!}
) or orthonormal rows (
a
a
∗
=
i
m
{\displaystyle aa^{*}=i_{m}\,\!}
), then:
a
+
=
a
∗
{\displaystyle a^{+}=a^{*}\,\!}
.
orthogonal projection matrices
if
a
{\displaystyle a\,\!}
orthogonal projection matrix, i.e.
a
=
a
∗
{\displaystyle a=a^{*}}
,
a
2
=
a
{\displaystyle a^{2}=a}
, pseudoinverse trivially coincides matrix itself:
a
+
=
a
{\displaystyle a^{+}=a\,\!}
.
circulant matrices
for circulant matrix
c
{\displaystyle c\,\!}
, singular value decomposition given fourier transform, singular values fourier coefficients. let
f
{\displaystyle {\mathcal {f}}}
discrete fourier transform (dft) matrix, then
c
=
f
⋅
Σ
⋅
f
∗
c
+
=
f
⋅
Σ
+
⋅
f
∗
{\displaystyle {\begin{aligned}c&={\mathcal {f}}\cdot \sigma \cdot {\mathcal {f}}^{*}\\c^{+}&={\mathcal {f}}\cdot \sigma ^{+}\cdot {\mathcal {f}}^{*}\end{aligned}}}
^ cite error: named reference ig2003 invoked never defined (see page).
^ stallings, w. t.; boullion, t. l. (1972). pseudoinverse of r-circulant matrix . proceedings of american mathematical society. 34: 385–388. doi:10.2307/2038377.
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