Generalised Inverse

Definition.

Given any matrix there exist a matrix (generalised inverse denoted by g-inv ). Such that whenever is a consistent system, ie
is a solution and dosen't depend on

Theorem.

Every matrix has a g-inv
Proof.

Suppose A be a matrix
if as the is trivial only consistent system is so the matrix suffices. In-fact any matrix works as g-inv let and as there exist a rank factorization let be a left inverse of and be right inverse of and let

Let be a consistent solution

Claim.

is solution of

Since

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Theorem.

THAE
i)
ii)
iii) for some matrix

Proof.

and the rank is same hence

as column of A is contained in there exist

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Theorem.

TFAE
i)
ii)
iii) for some D

Proof.

and the rank same hence

as every row in can we written as where is a column vector hence we can construct a matrix st.

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Theorem.

For any 2 matrices A and TFAE
i) is a g-inv of A
ii) AGA=A
iii)AG is idempotent and
iV) GA is idempotent and

Proof.

As is a consistent system for columns of A
so

Let be a consistent system ie, there exist a st.

so
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Corollary.

if is a g-inv of A then is a projector into along

is projector into along

Claim.

Proof.

to show the other way
let ie,

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Theorem.

Let be a g-inv of A then the system is consistent iff

Corollary.

Let be a g-inv of A then iff

Theorem.

a) If A is full column rank matrix, is a g-inv of A iff is a left inverse of A
b) If A is a full row rank matrix is a g-inv of A iff is a right inverse of A
c) If A is a Invertable is the unique of A

Proof.

Let be a left inverse of A
conversely let be a - inv so let be a left inverse of A then
so is left inverse of

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