1. Outline Properties on the Matrix Inverse Matrix Inverse Lemma
Left and Right Inverse
Moore-Penrose Inverse

2. Properties on the Matrix Inverse
If the inverse of n*n matrix A exists, then
The dimension of Ax is n;
span(A) = n
The dimension of x, for Ax = 0, is 0.
Zero space of A is null space
If A and B are invertible, then (AB)-1 = B-1A-1
Adjoint matrix: A-1 = adj(A) / det(A)

3. Matrix Inverse Lemma
n*n matrix A invertible, A plus a rank-one matrix (A + xyH) invertible, then
The following matrix inverse formula
Exercise !!!

4. Blockwise Matrix Inverse
If matrix A invertible
If matrices A and D are invertible
Applied when the inverse of A or D is only a scale

5. Left and Right Matrix Inverse
Inverse Matrix of A
A must be a square matrix, must be full rank
Other possibilities of A
A is not full rank??
A is not square??
“Inverse of A” in the above scenarios?

6. Left and Right Matrix Inverse
Matrix not square
Left inverse: GA= I
Right inverse: AG = I
Three possibilities
G exists and unique:
G exists but not unique:
G does not exist: G not of full rank
Left pseudo inverse: L = (AHA)-1AH
Right pseudo inverse: R = AH (AAH)-1

7. Left and Right Matrix Inverse
Left pseudo inverse: L = (ATA)-1AT
Least square solution for Ax = b
Right pseudo inverse: R = AT (AAT)-1
Minimum norm solution for for Ax = b

8. Moore-Penrose Inverse
Matrix A may not be full rank
MP Inverse A┼: AA┼A = A, A┼AA┼ = A┼,
AA┼ = (AA┼)H, A┼A = (A┼A)H
Motivations for the MP Inverse
The minimum norm solution Ax = b
Also the least square solution Ax = b

9. Properties on the Moore-Penrose Inverse
MP inverse is unique
Proof outline: consider another MP inverse B
Consider matrix: AA┼-AB
(AA┼-AB) (AA┼-AB) = 0, (AA┼-AB) is Hermitian
Then AA┼-AB = 0, and A┼A-BA = 0 in the same way
Thus A┼ = A┼AA┼ = BAB = B
Point：if for Hermitian matrix A, AA = 0,
then AAH = 0, and A = 0.

10. Properties on the Moore-Penrose Inverse
MP inverse of the conjugate transpose
(AT)┼ = (A┼)T, (AH)┼ = (A┼)H
Invertible matrix A: A┼ =A-1
MP inverse of MP inverse: (A┼) ┼ = A
MP inverse of Full-rank matrix
Full column rank: A┼ = (AHA)-1AH
Full row rank: A┼ = AH (AAH)-1
Non-singular square matrix: A┼ = A-1

11. Properties on the Moore-Penrose Inverse
MP inverse of vectors
Zero vector: x┼ = 0 for x = 0;
Nonzero vector: x┼ = xH/(xHx) for x ≠ 0
Orthonormal matrix
Inverse matrix: A┼ = A-1 = AH
MP inverse of the diagonal matrix
D = diag(d1, d2, …, dN), D┼ = diag(d1┼, d2┼, …, dN┼),
di┼ = di-1 for di ≠ 0, and di┼ = 0 for di = 0.