Mathcalc8 Basis : matrix calculation

Mathcalc8 focus more in list arithmetic, so we only provide 'essential' support for matrix calculation. In this post, we are going to do a lot of matrix calculation and first of all, we give a definition of matrix :

A p x q matrix A is defined as :

A = (a11,.....,a1j,.....,a1q)
    (.         .          .)
    (.         .          .)
    (ai1,.....,aij         .)
    (.                    .)
    (.                    .)
    (ap1,...............,apq)

where aij is the element of the matrix A at row ith and column jth

Matrix is treated as a list of list on Mathcalc8. This means that matrix A defined above is viewed as a list of p elements with each elment is itself a list with q elements. Therefore, create a matrix is the same as create a list which we should be very familiar now. As an example, we will show how to create a matrix (Note : Free version of Mathcalc8 do not support matrix calculation) :

Enter a 2 x 3 matrix a = (1, 2, 3)
                         (4, 5, 6) 

Input 1 : a=((1,2,3),(4,5,6))
Ans. set a = 1, 2, 3, 4, 5, 6
 

We can see that a is defined as a list of two elements. (1, 2, 3) is the first element and (4, 5, 6) is the second element. Both elements are also a list. To show the first row of a :

Input 2 : a(0)
Ans. 1, 2, 3
 

Mathcalc8 do not have a way to show individual element of a (input a(0)(1) do not work). So when we want to change one element of the matrix, say, change the first row, second column element from 2 to -1, we need to click the 'Dec' button to go back to Input 1 and make the change or we can use the following trick :

Input 3 : b=a(0)
Ans. set b = 1, 2, 3
 

Input 4 : b(1)=-1
Ans. set b = 1, -1, 3
 

Input 5 : a(0)=b
Ans. set a(0) = 1, -1, 3
 

Input 6 : a
Ans. 1, -1, 3, 4, 5, 6
 

We can see that b is used as an intermediate variable to make the change. Now, the original value of 2 in matrix a was changed to -1.

Next, we show how to get the transpose of matrix a through the function 'trans' :

Input 7 : b=trans(a)
Ans. set b = 1, 4, -1, 5, 3, 6
 

The first row of b is :

Input 8 : b(0)
Ans. 1, 4
 

The second row of b is :

Input 9 : b(1)
Ans. -1, 5
 

The third row of b is :

Input 10 : b(2)
Ans. 3, 6
 

So, we can see that b is a 3 x 2 matrix.

One precaution about matrix definition is on the definition of n x 1 matrix. For example, if we want to define a 3 x 1 matrix as follow :

a = (1)
    (2)
    (3) 

We should not define it by setting a=((1),(2),(3))! Instead, we need to define the matrix using the function 'trans' as follow :

Input 11 : a=trans((1,2,3))
Ans. set a = 1, 2, 3
 

Input 12 : a(0)
Ans. 1
 

Next, we would show how to do matrix arithmetic. They are similar to list arithmetic except that there is two kind of multiplication for matrix :
1. Addition and Subtraction uses the same symbol '+' and '-' :

Add (1, 2) + (5, 6)
    (3, 4)   (7, 8) 

Input 13 : ((1,2),(3,4))+((5,6),(7,8))
Ans. 6, 8, 10, 12
 

Subtract (1, 2) - (5, 6)
         (3, 4)   (7, 8) 

Input 14 : ((1,2),(3,4))-((5,6),(7,8))
Ans. -4, -4, -4, -4
 

Add 1 + (5, 6)
        (7, 8) 

Input 15 : 1+((5,6),(7,8))
Ans. 6, 7, 8, 9
 

Subtract 1 - (5, 6)
             (7, 8) 

Input 16 : 1-((5,6),(7,8))
Ans. -4, -5, -6, -7
 

2. Scale multiplication and division also uses same symbol '*' and '/' :

Multiply 2 x (1, 2)
             (3, 4) 

Input 17 : 2*((1,2),(3,4))
Ans. 2, 4, 6, 8
 

Division  (1, 2) / 2
          (3, 4) 

Input 18 : ((1,2),(3,4))/2
Ans. 0.5, 1, 1.5, 2
 

3. Matrix-matrix multiplication, we need to use function 'mmult' for standard matrix multiply :

Matrix Multiply (1, 2) x (5, 6)
                (3, 4)   (7, 8) 

Input 19 : mmult(((1,2),(3,4)),((5,6),(7,8)))
Ans. 19, 22, 43, 50
 

If we use '*' to do matrix multiplication, the multiplication would be done on the same position of elements between the two matrix :

Multiply (1, 2) x (5, 6)
         (3, 4)   (7, 8) 

Input 20 : ((1,2),(3,4))*((5,6),(7,8))
Ans. 5, 12, 21, 32
 

Matrix Multiply (2,-6) x (5)
                         (3) 

Input 21 : mmult((2,-6),trans((5,3)))
Ans. -8
 

Matrix Multiply (5) x (2,-6)
                (3) 

Input 22 : mmult(trans((5,3)),(2,-6))
Ans. 10, -30, 6, -18
 

Matrix Multiply (2, 1) x (-1, 6)
                (4, 3)   ( 3, 2)
                (1, 2) 

Input 23 : mmult(((2,1),(4,3),(1,2)),((-1,6),(3,2)))
Ans. 1, 14, 5, 30, 5, 10
 

Next, we define two matrix a and b as follow :

a = (-2 , 4, 3+2i)        b = (5, 1,  -4)
    (1  , 6,   -7)            (1, 3, -3i) 

Input 24 : a=((-2,4,3+2*sqrt(-1)),(1,6,-7))
 

Input 25 : b=((5,1,-4),(1,3,-3*sqrt(-1)))
 

1. Evaluate 2a - 3b :

Input 26 : 2*a-3*b
Ans. 19, 5, 18+4i, -1, 3, -14+9i
 

2. Evaluate ab^T. The symbol 'T' means 'transpose' :

(-2 , 4, 3+2i) x (5,   1)
(1  , 6,   -7)   (1,   3)
                 (-4,-3i) 

Input 27 : mmult(a,trans(b))
Ans. -18-8i, 16-9i, 39, 19+21i
 

3. Evaluate ab^H. The symbol 'H' means 'Hermitian transpose', (i.e. the matrix is being transposed and then take complex conjugate for all its elements). To take complex conjugate for a complex number means that we change the sign of the complex number's imaginary part with the real part kept fixed. In Mathcalc8, this is done by using the function 'conj' :

(-2 , 4, 3+2i) x (5,   1)
(1  , 6,   -7)   (1,   3)
                 (-4, 3i) 

Input 28 : mmult(a,conj(trans(b)))
Ans. -18-8i, 4+9i, 39, 19-21i
 

4. Evaluate a^Tb :

(-2,   1) x (5, 1,  -4)
(4,    6)   (1, 3, -3i)
(3+2i,-7)

Input 29 : mmult(trans(a),b)
Ans. -9, 1, 8-3i, 26, 22, -16-18i, 8+10i, -18+2i, -12+13i
 

5. Evaluate a^Hb :

(-2,   1) x (5, 1,  -4)
(4,    6)   (1, 3, -3i)
(3-2i,-7) 

Input 30 : mmult(conj(trans(a)),b)
Ans. -9, 1, 8-3i, 26, 22, -16-18i, 8+10i, -18-2i, -12+29i
 

6. Evaluate trace(ab^H). The trace of a matrix a is equal to the summation of all of its diagonal elements a_ii :

Input 31 : c=mmult(a,conj(trans(b)))
 

Input 32 : tr=0
Ans. set tr = 0
 

Input 33 : table('s=c(x)|tr=tr+s(x)',id(2))
 

Input 34 : tr
Ans. 1-29i
 

7. Evaluate (ab^H)⁵ :

Input 35 : y=c
 

Input 36 : table('y=mmult(c,y)',id(4))
 

Input 37 : y
Ans. 13935162-1400900i, -603116-5405403i, -2052971-6510816i, -7712021-734589i