Generalized Inverse

Question 1

Let A be an mxn matrix. A⁺ is called the Moore-Penrose inverse if it satisfies the following conditions:

Answer 1

• AA⁺ is symmetric.
• A⁺A is symmetric.
• AA⁺A = A
• A⁺AA⁺ = A⁺

Question 2

Let X be an nxp matrix with rank p. X⁺ =

Answer 2

X⁺ = (X’X)^-1X’

Question 3

Every matrix has a Moore-Penrose inverse and it is

Answer 3

unique.

Question 4

The Moore-Penrose inverse of A’, ie (A’)⁺=

Answer 4

(A’)⁺ = (A⁺)’

Question 5

The Moore-Penrose inverse of A⁺ =

Answer 5

A

Question 6

If the rank of A is r, then each of the following matrices also have rank r (Moore-Penrose inverse combinations)

Answer 6

A⁺, AA⁺, A⁺A

Question 7

If A is non singlular, then A⁺ =

Answer 7

A^-1

Question 8

If A is symmetric idempotent, then A⁺ =

Answer 8

A

Question 9

The matrices AA⁺, A⁺A, I-AA⁺, I-A⁺A are all

Answer 9

symmetric idempotent

Question 10

For any matrix A, (A’A)⁺ =

Answer 10

A⁺(A’)⁺

Question 11

Let P be an mxm orthonormal matrix, let Q be an nxn orthonormal matrix, and let A be any mxn matrix. (PAQ)⁺ =

Answer 11

Q’A⁺P’

Question 12

Let X = [X₁ X₂],

XX⁺X₁ =

Answer 12

X₁

Question 13

Let A be an mxn matrix. A^- is called a generalized inverse of A if

Answer 13

AA^-A = A

Question 14

Let X be an mxn matrix with rank r>0. The following conditions hold wrt generalized inverse:

Answer 14

• rank of A^- is not less than rank of A
• XX^- and X^-X are idempotent
• the ranks of XX^- and X^-X are r
• if X^- is any generalized inverse of X, then (X^-)’ is a generalized inverse of X’
• X^-X = I IFF X has full column rank
• XX^- = I IFF X has full row rank

Question 15

K = ____ is invariant to the choice of generalized inverse?

Answer 15

K = X(X’X)^-X’

Question 16

X(X’X)^-X’ = in terms of Moore-Penrose inverse?

Answer 16

X(X’X)^-X’ = XX⁺

Question 17

The following conditions hold for K = X(X’X)^-​X’

Answer 17

• K is symmetric idempotent
• rank of K is r, the rank of X
• KX =X and X’K =X’
• (X’X)^-X’ is a generalized inverse of X for any generalized inverse of X’X
• X(X’X)^- is a generalized inverse of X’ for any generalized inverse of X’X

Question 18

If A is an nxn non-singlular matrix, then the system Ax = y has ___ solutions

Answer 18

Ax=y has a single unique solution

Question 19

The system Ax = y has a solution IFF y is

Answer 19

Ax = y has a solution IFF y is in the column space of A

Question 20

The system Ax = y has a solution IFF (inverse)

Answer 20

Ax = y has a solution IFF AA^-y = y

Question 21

Let A be an mxn matrix. The system Ax = 0 has a solution other than x = 0 IFF the rank of A

Answer 21

IFF the rank of A < n

Question 22

Let A be an mxn matrix and suppose that a solution exists to the system Ax = y. For each nx1 vector z, the vector x₀ is a solution where

Answer 22

x₀ = A^-y + (I - A^-A)z