MoorePenrose Inverse of a Matrix
(E.H. Moore (1920), Arne Bjerhammar and Roger Penrose (1955))
Programs for the HP Prime and Casio Prizm (can be used for fx9860g II and fx9750g II) included
Also known as a pseudoinverse, the MoorePenrose inverse of a matrix, denoted by A^+ (capital A with a supersubscript of a plus sign), is an inverse of matrix. Different from the "true" matrix inverse, the MoorePenrose inverse allows for nonsquare matrices. Primarily, the MoorePenrose inverses are calculated is assist in solving linear leastsquare equations.
Properties: MoorePenrose Inverse
A A^+ A = A
A^+ A A^+ = A^+
(A A^+)^T = A
(A^+ A)^T = A^+
A^+ is unique
Calculation
One method of calculating the MoorePenrose Inverse is the following:
For matrix A of size m × n:
Let r = row rank of A (number of linearly independent rows)
Let c = column rank of A (number of linearly independent columns)
Usually r = c
The matrix rank, denoted rank(A) = min(r, c)
If r ≤ c and rank(A) = r,
Then A^+ = A^T (A A^T)^1
If r ≥ c and rank(A) = c,
Then A^+ = (A^T A)^1 A^T
The programs listed on this blog entry assume the first conditions of each of the test are met, since usually, r = c.
Sources
Schott, James R. "Matrix Analysis for Statistics: 2nd Edition". Wiley & Sons. Hoboken, NJ. 2005
Wikipedia article, retrieved 8/29/2014  URL:
http://en.m.wikipedia.org/wiki/Moore–Penrose_pseudoinverse
HP Prime Program MPINVERT
Input: MPINVERT(matrix). Execute this from Home or CAS screen.
Program:
EXPORT MPINVERT(mtx)
BEGIN
// 20140827 EWS
// MoorePenrose Matrix Inverse
LOCAL r,c,d,n;
d:=SIZE(mtx);
r:=d(1);
c:=d(2);
n:=RANK(mtx);
CASE
IF n==c THEN
RETURN (TRN(mtx)*mtx)^1*mtx;
END;
IF n==r THEN
RETURN TRN(mtx)*(mtx*TRN(mtx))^1;
END;
DEFAULT
RETURN "No Solution Found";
END;
END;
Casio Prizm Program MPINVERT
The program prompts for the matrix. Since the Prizm does not have a matrix rank command, it is determined by use of the rref command and pair of For loops. This program should work verbatim on the fx9860g II and fx9750g II.
The matrix is stored in Mat X. List 1 and Mat Y are used for calculation. The result is stored in Mat Z.
If you are using a TI84, you would probably use an approach similar to what is presented:
Program MPINVERT
"MOOREPENROSE INVERSE"
"EWS 20140827"
"MATRIX X"? → Mat X
Dim Mat X→List 1
List 1[1] → R
List 1[2] → C
Rref Mat X → Mat Y
0 → T
For 1 → I To R
For 1 → J To C
If Mat Y[I, J] ≠ 0
Then
T + 1 → T
Break
IfEnd
Next
Next
If T = R
Then
Trn Mat X * (Mat X * Trn Mat X) ^ 1 → Mat Z
"Mat Z"
Mat Z ◢ // display
Stop
IfEnd
If T = C
Then
(Trn Mat X * Mat X) ^ 1 * Trn Mat X → Mat Z
"Mat Z"
Mat Z ◢ // display
Stop
IfEnd
"NO INVERSE"
Examples
Matrix:
[ [ 1, 2, 3 ] [ 3, 4, 0 ] ]
MoorePenrose Inverse:
[ [ 8/229, 31/229 ] [ 6/229, 34/229 ] [ 75/229, 33/229 ] ]
Matrix:
[ [7, 4, 6, 7] [1, 5, 3, 3] ]
MoorePenrose Inverse: (to four decimal places)
[ [0.0489, 0.0338] [0.0194, 0.1092] [0.0360, 0.0600] [0.0520, 0.0800] ]
This blog is property of Edward Shore. 2014
A blog is that is all about mathematics and calculators, two of my passions in life.
Friday, August 29, 2014
MoorePenrose Inverse of a Matrix
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