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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

Sums

Products

Powers

Examples

Description

• 

The result that is returned for a particular expression is described below. In each of the following sections:

  

- A is an Array

  

- M is a Matrix

  

- V is a Vector

  

- 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

 

 

     

The component-wise sum of  and ,

 

if the dimensions match;

 

otherwise, returns an error

     

Adds  to every element of

     

Returns unevaluated

 

 

     

The component-wise sum of  and ,

 

if the dimensions match;

 

otherwise, returns an error

     

Adds  to the main diagonal of

     

Returns unevaluated

 

 

     

The component-wise sum of  and ,

 

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. The . operator performs noncommutative or dot product multiplcation. For more information, see dot.

  

 

     Expression

Result

 

 

     

The component-wise product of  and , if the dimensions match;

 

otherwise, returns an error

     

Multiplies every element of  by

     

Multiplies every element of  by

 

 

     

An error (must use the '.' (dot) operator)

     

Multiplies every element of  by

     

Multiplies every element of  by

 

 

     

An error (must use the '.' (dot) operator)

     

Multiplies every element of  by

     

Multiplies every element of  by

 

 

     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.

  

There are two cases in which the exponent is interpreted specially:  and .  (The deprecated notations,  and , respectively, are similarly interpreted.)

  

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.

     

The component-wise exponentiation of .

 

 

     

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)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

The * operator cannot be used to multiply Matrices or Vectors.

M1 * M2;

Error, (in `rtable/Product`) use *~ for elementwise multiplication of Vectors or Matrices; use . (dot) for Vector/Matrix multiplication

(15)

(16)

Note the difference between * and . when one operand is a symbolic scalar.

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

See Also

Array

dot

LinearAlgebra[Transpose]

Matrix

rtable

Vector

 


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