Example 1: Analyze the isometry given by the matrix

 

 

 

 

 

 

 

 

 

 

 

 

 

0

–1

 

 

 

 

 

 

 

 

 

–1

0

 

 

 

 

 

 

 

Command:

 

 

 

 

 

 

 

 

 

ISOM([[0,-1] [-1,0]])

 

 

 

 

Result:

{ [1, 1] –1}, meaning the matrix represents a symmetry in the line y = –x, and this is an indirect

 

isometry.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

– 3

 

 

 

 

 

 

 

 

--

---------

 

 

 

 

 

 

 

 

2

2

 

 

 

 

 

 

 

 

 

3

1

 

 

 

 

 

 

------

--

 

 

Example 2:

Analyze the isometry given by the matrix

 

 

2

2

 

 

 

 

 

 

 

Command:

ISOM([[1/2, -√3/2][√3/2, 1/2]])

 

 

 

 

Result:

{ π/3, 1 }, meaning the matrix represents a rotation of π/3 radians, and this is a direct isometry.

See also:

MKISOM

 

 

 

 

ISPRIME?

 

 

 

 

 

 

 

 

 

 

Type:

Function

 

 

 

 

Description:

Tests if a number is prime. For numbers of the order of 1014 or greater (to be exact, greater than

 

341550071728321), tests if the number is a pseudoprime; this has a chance of less than 1 in 1012

 

of wrongly identifying a number as a prime.

 

 

 

Access:

PARITH or Arithmetic, !ÞINTEGER L

Input:

An object that evaluates to an integer or a whole real number.

Output:

1 (True) if the number is prime, 0 (False) if it is not.

Flags:

Exact mode must be set (flag –105 clear).

 

 

 

 

 

Numeric mode must not be set (flag –3 clear).

 

 

 

See also:

NEXTPRIME, PREVPRIME

 

 

 

 

I→R

Function

 

 

 

 

Type:

 

 

 

 

Description:

Converts an integer into a real number.

 

 

 

 

Access:

…Ú REWRITE

 

 

 

 

Flags:

Exact mode must be set (flag –105 clear). Numeric mode must not be set (flag –3 clear). The flags

 

affect the output only if the input is not an integer.

Input:

Level 1/Argument 1: An integer or real number.

Output:

Level 1/Item 1: The integer converted to a real number.

See also:

→NUM, R→I, XNUM

 

 

 

 

JORDAN

 

 

 

 

 

 

 

 

 

 

Type:

Command

 

 

 

 

Description:

Diagonalization, or Jordan cycle decomposition, of a matrix. Computes the eigenvalues,

 

eigenvectors, minimum polynomial, and characteristic polynomial of a matrix.

Access:

Matrices, !Ø LEIGENVECTORS

 

 

 

 

Input:

An n × n matrix.

 

 

 

 

Output:

Level 4/Item 1: The minimum polynomial.

 

 

 

 

 

Level 3/Item 2: The characteristic polynomial.

 

 

 

 

Level 2/Item 3: A list of characteristic spaces tagged by the corresponding eigenvalue (either a

3-122 Full Command and Function Reference