What is moore penrose inverse of a matrix for a 2x2 matrix example
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K will denote one of the fields of real or complex numbers, denoted {\displaystyle \mathbb {R} } \mathbb {R} , {\displaystyle \mathbb {C} } \mathbb {C} , respectively. The vector space of {\displaystyle m\times n} m\times n matrices over {\displaystyle K} K is denoted by {\displaystyle K^{m\times n}} K^{m\times n}.
For {\displaystyle A\in K^{m\times n}} {\displaystyle A\in K^{m\times n}}, {\displaystyle A^{\mathrm {T} }} A^{\mathrm {T} } and {\displaystyle A^{*}} A^{*} denote the transpose and Hermitian transpose (also called conjugate transpose) respectively. If {\displaystyle K=\mathbb {R} } K=\mathbb {R} , then {\displaystyle A^{*}=A^{\mathrm {T} }} A^{*}=A^{\mathrm {T} }.
For {\displaystyle A\in K^{m\times n}} {\displaystyle A\in K^{m\times n}}, {\displaystyle \operatorname {im} (A)} \operatorname {im} (A) denotes the range (image) of {\displaystyle A} A (the space spanned by the column vectors of {\displaystyle A} A) and {\displaystyle \operatorname {ker} (A)} \operatorname {ker} (A) denotes the kernel (null space) of {\displaystyle A} A.
Finally, for any positive integer {\displaystyle n} n, {\displaystyle I_{n}\in K^{n\times n}} {\displaystyle I_{n}\in K^{n\times n}} denotes the {\displaystyle n\times n} n\times n identity matrix.
Definition
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Linear least-squares
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See also: Linear least squares (mathematics)
The pseudoinverse provides a least squares solution to a system of linear equations.[23] For {\displaystyle A\in K^{m\times n}\,\!} {\displaystyle A\in K^{m\times n}\,\!}, given a system of linear equations
{\displaystyle Ax=b,\,} Ax=b,\,
in general, a vector {\displaystyle x} x that solves the system may not exist, or if one does exist, it may not be unique. The pseudoinverse solves the "least-squares" problem as follows:
{\displaystyle \forall x\in K^{n}\,\!} {\displaystyle \forall x\in K^{n}\,\!}, we have {\displaystyle \|Ax-b\|_{2}\geq \|Az-b\|_{2}} {\displaystyle \|Ax-b\|_{2}\geq \|Az-b\|_{2}} where {\displaystyle z=A^{+}b} z=A^{+}b and {\displaystyle \|\cdot \|_{2}} \|\cdot \|_{2} denotes the Euclidean norm. This weak inequality holds with equality if and only if {\displaystyle x=A^{+}b+\left(I-A^{+}A\right)w} {\displaystyle x=A^{+}b+\left(I-A^{+}A\right)w} for any vector w; this provides an infinitude of minimizing solutions unless A has full column rank, in which case {\displaystyle \left(I-A^{+}A\right)} {\displaystyle \left(I-A^{+}A\right)} is a zero matrix.[24] The solution with minimum Euclidean norm is {\displaystyle z.} z.[24]
This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let {\displaystyle B\in K^{m\times p}} {\displaystyle B\in K^{m\times p}}.
{\displaystyle \forall X\in K^{n\times p}\,\!} {\displaystyle \forall X\in K^{n\times p}\,\!}, we have {\displaystyle \|AX-B\|_{\mathrm {F} }\geq \|AZ-B\|_{\mathrm {F} }} \|AX-B\|_{\mathrm {F} }\geq \|AZ-B\|_{\mathrm {F} } where {\displaystyle Z=A^{+}B} Z=A^{+}B and {\displaystyle \|\cdot \|_{\mathrm {F} }} \|\cdot \|_{\mathrm {F} } denotes the