|
Algebra with Matrices, Vectors, and Arrays
|
|
Expressions involving sums, products, and powers with rtable objects are evaluated directly. An rtable object is either an Array, Matrix, or Vector.
|
|
|
Description
|
|
|
•
|
The result that is returned for a particular expression is described below. In each of the following sections:
|
|
|
- c is a numeric constant
|
|
|
- s is a non-numeric scalar
|
|
|
|
Sums
|
|
|
|
The result that is returned when an expression of type '+' includes at least one rtable depends on the operands.
|
|
Expression
|
Result
|
|
|
|
|
![A[1]+A[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math63.png)
|
The component-wise sum of ![A[1]](/support/helpjp/helpview.aspx?si=9025/file00881/math67.png) and ![A[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math69.png) ,
|
|
|
if the dimensions match;
|
|
|
otherwise, returns an error
|
|
|
Adds  to every element of 
|
|
|
Returns unevaluated
|
|
|
|
|
![M[1]+M[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math115.png)
|
The component-wise sum of ![M[1]](/support/helpjp/helpview.aspx?si=9025/file00881/math119.png) and ![M[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math121.png) ,
|
|
|
if the dimensions match;
|
|
|
otherwise, returns an error
|
|
|
Adds  to the main diagonal of 
|
|
|
Returns unevaluated
|
|
|
|
|
![V[1]+V[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math167.png)
|
The component-wise sum of ![V[1]](/support/helpjp/helpview.aspx?si=9025/file00881/math171.png) and ![V[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math173.png) ,
|
|
|
if dimensions and orientations match;
|
|
|
otherwise, returns an error
|
|
|
An error
|
|
|
Returns unevaluated
|
|
|
|
|
other
|
All other combinations raise errors
|
|
|
|
|
Direct evaluation of these expressions is implemented by calls to the rtable/Sum library routine.
|
|
|
|
Products
|
|
|
|
The result that is returned when an expression of type '*' includes at least one rtable depends on the operands. If the operands are either Matrices, Vectors or a combination of each (with appropriate dimensions), the '.' operator must be used. For more information, see dot.
|
|
Expression
|
Result
|
|
|
|
|
![A[1]*A[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math260.png)
|
The component-wise product of ![A[1]](/support/helpjp/helpview.aspx?si=9025/file00881/math264.png) and ![A[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math266.png) , if the dimensions match;
|
|
|
otherwise, returns an error
|
|
|
Multiplies every element of  by 
|
|
|
Returns unevaluated
|
|
|
|
|
![M[1]*M[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math305.png)
|
An error (must use the '.' (dot) operator)
|
|
|
Multiplies every element of  by 
|
|
|
Returns unevaluated
|
|
|
|
|
![V[1]*V[2]](/support/helpjp/helpview.aspx?si=9025/file00881/math339.png)
|
An error (must use the '.' (dot) operator)
|
|
|
Multiplies every element of  by 
|
|
|
Returns unevaluated
|
|
|
|
|
other
|
All other combinations raise errors
|
|
|
|
|
Direct evaluation of these expressions is implemented by calls to the rtable/Product library routine.
|
|
|
|
Powers
|
|
|
|
The result that is returned when an expression of type '^' includes an rtable object base depends on the exponent type.
|
|
|
Otherwise, the following interpretations of a power of an rtable apply.
|
|
Expression
|
Result
|
|
|
|
|
|
The component-wise exponentiation of  .
|
|
|
Constant c can be any (complex) numeric value.
|
|
|
Returns unevaluated
|
|
|
|
|
|
If M is square and c is a positive integer, then
|
|
|
the result is the matrix product  ... 
|
|
|
(c factors using the dot operator)
|
|
|
If c is a negative integer, the result is
|
|
|
 (  ...  ) ( factors using
|
|
|
the dot operator)
|
|
|
If  , the result is 
|
|
|
If  is not an integer, it returns unevaluated
|
|
|
Returns unevaluated
|
|
|
|
|
Direct evaluation of these expressions is implemented by calls to the rtable/Power library routine.
|
|
|
|
Examples
|
|
|
>
|
|
|
| (1) |
|
>
|
![A1 := Array([[1, 2], [3, 4]])](/support/helpjp/helpview.aspx?si=9025/file00881/math550.png)
|
|
![A1 := Array([[1, 2], [3, 4]])](/support/helpjp/helpview.aspx?si=9025/file00881/math553.png)
| (2) |
|
>
|
![A2 := Array([[u, v], [w, x]])](/support/helpjp/helpview.aspx?si=9025/file00881/math557.png)
|
|
![A2 := Array([[u, v], [w, x]])](/support/helpjp/helpview.aspx?si=9025/file00881/math560.png)
| (3) |
|
>
|
|
|
![M1 := Matrix([[1, -2], [2, 1]])](/support/helpjp/helpview.aspx?si=9025/file00881/math567.png)
| (4) |
|
>
|
|
|
![M2 := Matrix([[2, 4], [3, 4]])](/support/helpjp/helpview.aspx?si=9025/file00881/math574.png)
| (5) |
|
>
|
|
|
](/support/helpjp/helpview.aspx?si=9025/file00881/math581.png)
| (6) |
|
>
|
|
|
![Array([[1+u, 2+v], [3+w, 4+x]])](/support/helpjp/helpview.aspx?si=9025/file00881/math588.png)
| (7) |
|
>
|
|
|
![Array([[3, 4], [5, 6]])](/support/helpjp/helpview.aspx?si=9025/file00881/math595.png)
| (8) |
|
>
|
|
|
![s+Array([[1, 2], [3, 4]])](/support/helpjp/helpview.aspx?si=9025/file00881/math602.png)
| (9) |
|
>
|
|
|
![Matrix([[3, 2], [5, 5]])](/support/helpjp/helpview.aspx?si=9025/file00881/math609.png)
| (10) |
|
>
|
|
|
![Matrix([[3, -2], [2, 3]])](/support/helpjp/helpview.aspx?si=9025/file00881/math616.png)
| (11) |
|
>
|
|
|
![s+Matrix([[1, -2], [2, 1]])](/support/helpjp/helpview.aspx?si=9025/file00881/math623.png)
| (12) |
|
>
|
|
|
![Array([[u, 2*v], [3*w, 4*x]])](/support/helpjp/helpview.aspx?si=9025/file00881/math630.png)
| (13) |
|
>
|
|
|
![Array([[u, 2*v], [3*w, 4*x]])](/support/helpjp/helpview.aspx?si=9025/file00881/math637.png)
| (14) |
The * operator cannot be used to multiply Matrices or Vectors.
|
>
|
|
|
>
|
|
|
![Matrix([[2, -4], [4, 2]])](/support/helpjp/helpview.aspx?si=9025/file00881/math658.png)
| (15) |
|
>
|
|
|
![Matrix([[2, -4], [4, 2]])](/support/helpjp/helpview.aspx?si=9025/file00881/math665.png)
| (16) |
Note the difference between * and . when one operand is a symbolic scalar.
|
>
|
|
|
![Matrix([[s, -2*s], [2*s, s]])](/support/helpjp/helpview.aspx?si=9025/file00881/math674.png)
| (17) |
|
>
|
|
|
![s.Matrix([[1, -2], [2, 1]])](/support/helpjp/helpview.aspx?si=9025/file00881/math681.png)
| (18) |
|
>
|
|
|
![Matrix([[-4, -4], [7, 12]])](/support/helpjp/helpview.aspx?si=9025/file00881/math688.png)
| (19) |
|
>
|
|
|
](/support/helpjp/helpview.aspx?si=9025/file00881/math695.png)
| (20) |
|
>
|
|
|
![Array([[1, 4], [9, 16]])](/support/helpjp/helpview.aspx?si=9025/file00881/math702.png)
| (21) |
|
>
|
|
|
![Array([[1, 2^s], [3^s, 4^s]])](/support/helpjp/helpview.aspx?si=9025/file00881/math709.png)
| (22) |
|
>
|
|
|
![Matrix([[16, 24], [18, 28]])](/support/helpjp/helpview.aspx?si=9025/file00881/math716.png)
| (23) |
|
>
|
|
|
![Matrix([[2, 4], [3, 4]])^s](/support/helpjp/helpview.aspx?si=9025/file00881/math723.png)
| (24) |
|
>
|
|
|
![Matrix([[2, 3], [4, 4]])](/support/helpjp/helpview.aspx?si=9025/file00881/math730.png)
| (25) |
|
>
|
|
|
], [conjugate(y)]])](/support/helpjp/helpview.aspx?si=9025/file00881/math737.png)
| (26) |
|
|
