FMxN

An abstract M by N matrix space over a ring. Matrix spaces describe linear maps between vector spaces relative to the vector spaces' assumed bases. Matrix addition associates and commutes, and scalar multiplication associates, commutes, and distributes over matrix addition and scalar addition. Matrix and vector multiplication also both associate and distribute over matrix addition. Vectors in the row space multiply as columns on the right, and vectors in the column space multiply as rows on the left. Addition and scalar multiplication both have an identity element, and every matrix has an additive inverse. To the extent practicable, the following axioms should hold.

Axioms for matrix addition:

if 𝐀 and 𝐁 are matrices in this, then their sum 𝐀 + 𝐁 is also a matrix in this.
𝐀 + 𝐁 == 𝐁 + 𝐀 for all matrices 𝐀, 𝐁 in this.
(𝐀 + 𝐁) + 𝐂 == 𝐀 + (𝐁 + 𝐂) for all matrices 𝐀, 𝐁, 𝐂 in this.
this has a matrix zero such that zero + 𝐀 == 𝐀 for every matrix 𝐀 in this.
to every matrix 𝐀 in this corresponds a matrix -𝐀 in this such that 𝐀 + (-𝐀) == zero.

Axioms for scalar multiplication:

if 𝑥 is a scalar in this and 𝐀 is a matrix in this, then their product 𝑥 *: 𝐀 is also a matrix in this.
(𝑥 * 𝑦) *: 𝐀 == 𝑥 *: (𝑦 *: 𝐀) for all scalars 𝑥, 𝑦 and every matrix 𝐀 in this.
Scalar has an element unit such that unit *: 𝐀 == 𝐀 for every matrix 𝐀 in this.

Axioms for vector multiplication:

If 𝐀 is a matrix in this and 𝐯 is a vector in the row space of 𝐀, then their product 𝐀 :⋅ 𝐯 is a vector in the column space of 𝐀.
(𝐀 ⋅ 𝐁) :⋅ 𝐯 == 𝐀 :⋅ (𝐁 :⋅ 𝐯) for all matrices 𝐀, 𝐁 and every vector 𝐯 in the row space of 𝐁, where the row space of 𝐀 equals the column space of 𝐁.
𝐀 :⋅ 𝐯 == 𝐯 ⋅: 𝐀.T for every matrix 𝐀 and every vector 𝐯 in the row space of 𝐀.

Axioms for matrix multiplication:

if 𝐀 and 𝐁 are matrices and the row space of 𝐀 equals the column space of 𝐁, then the matrix product 𝐀 ⋅ 𝐁 exists.
(𝐀 ⋅ 𝐁) ⋅ 𝐂 == 𝐀 ⋅ (𝐁 ⋅ 𝐂) for all matrices 𝐀, 𝐁, 𝐂 where the row space of 𝐀 equals the column space of 𝐁, and the row space of 𝐁 equals the column space of 𝐂.
(𝐀 ⋅ 𝐁).T == 𝐁.T ⋅ 𝐀.T for all matrices 𝐀, 𝐁 where the row space of 𝐀 equals the column space of 𝐁.

Distributive laws:

𝑥 *: (𝐀 + 𝐁) == (𝑥 *: 𝐀) + (𝑥 *: 𝐁) for every scalar 𝑥 and all matrices 𝐀, 𝐁 in this.
(𝑥 + 𝑦) *: 𝐀 == (𝑥 *: 𝐀) + (𝑦 *: 𝐀) for all scalars 𝑥, 𝑦 and every matrix 𝐀 in this.
𝐯 ⋅: (𝐀 + 𝐁) == (𝐯 ⋅: 𝐀) + (𝐯 ⋅: 𝐁) for all matrices 𝐀, 𝐁 in the same matrix space, and every vector 𝐯 in the column space of 𝐀 and 𝐁.
(𝐀 + 𝐁) :⋅ 𝐯 == (𝐀 :⋅ 𝐯) + (𝐁 :⋅ 𝐯) for all matrices 𝐀, 𝐁 in the same matrix space, and every vector 𝐯 in the row space of 𝐀 and 𝐁.
𝐀 ⋅ (𝐁 + 𝐂) == (𝐀 ⋅ 𝐁) + (𝐀 ⋅ 𝐂) for all matrices 𝐁, 𝐂 in the same matrix space, and every matrix 𝐀 whose row space equals the colum space of 𝐁 and 𝐂.
(𝐀 + 𝐁) ⋅ 𝐂 == (𝐀 ⋅ 𝐂) + (𝐁 ⋅ 𝐂) for all matrices 𝐀, 𝐁 in the same matrix space, and every matrix 𝐂 whose column space equals the row space of 𝐀 and 𝐁.

Self Type: FMxN
Source: FMxN.scala
Version: 0.1
Since: 0.0

Linear Supertypes

VectorSpace, AnyRef, Any

Known Subclasses

F2x2, F3x3, F4x4, R2x2, R3x3, R4x4, RMxN, Real

Type Hierarchy Learn more about scaladoc diagrams

Type Members

type Col = (Col)#Vector

The type of vectors in the column space.
abstract type Matrix <: MatrixFMxN

The type of matrices in this matrix space.
trait MatrixFMxN extends VectorElement

A matrix in this matrix space.
type Row = (Row)#Vector

The type of vectors in the row space.
type Scalar = Element

The type of scalars in this matrix space.
The type of scalars in this matrix space.

Definition Classes
VectorSpace
type Transpose = (Transpose)#Matrix

The type of matrix transposes.
type Vector = Matrix

The type of vectors in this matrix space; equivalent to the type of matrices.
The type of vectors in this matrix space; equivalent to the type of matrices.

Definition Classes
FMxN → VectorSpace
trait VectorElement extends Any

A vector in this vector space.
A vector in this vector space.

Definition Classes
VectorSpace

Abstract Value Members

abstract val Col: FN { val Scalar: FMxN.this.Scalar.type }
abstract val Row: FN { val Scalar: FMxN.this.Scalar.type }

Returns the row space.
abstract val Scalar: Ring

Returns the scalar set of this vector space.
Returns the scalar set of this vector space.

Definition Classes
VectorSpace
implicit abstract def ScalarTag: ClassTag[Scalar]
abstract val Transpose: FMxN { ... /* 3 definitions in type refinement */ }

Returns the transpose of this matrix space.
abstract def apply(entries: Array[Scalar]): Matrix

Returns a new matrix with the given row-major entries.

Concrete Value Members

final def !=(arg0: Any): Boolean

Definition Classes
AnyRef → Any
final def ##(): Int

Definition Classes
AnyRef → Any
final def ==(arg0: Any): Boolean

Definition Classes
AnyRef → Any
final def asInstanceOf[T0]: T0

Definition Classes
Any
def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( ... )
def cols(cols: Col*): Matrix

Returns a new matrix with the given columns.
def dim: Int
final def eq(arg0: AnyRef): Boolean

Definition Classes
AnyRef
def equals(arg0: Any): Boolean

Definition Classes
AnyRef → Any
def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
final def getClass(): Class[_]

Definition Classes
AnyRef → Any
def hashCode(): Int

Definition Classes
AnyRef → Any
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
final def ne(arg0: AnyRef): Boolean

Definition Classes
AnyRef
final def notify(): Unit

Definition Classes
AnyRef
final def notifyAll(): Unit

Definition Classes
AnyRef
def product[A <: FMxN { val Scalar: FMxN.this.Scalar.type }, B <: FMxN { val Scalar: FMxN.this.Scalar.type }](matrixA: product.A.MatrixFMxN, matrixB: product.B.MatrixFMxN): Matrix

Returns the matrix product of the first matrix, whose column space equals this column space, times the second matrix, whose row space equals this row space, where the row space of the first matrix equals the column space of the second matrix.
def rows(rows: Row*): Matrix

Returns a new matrix with the given rows.
final def synchronized[T0](arg0: ⇒ T0): T0

Definition Classes
AnyRef
def toString(): String

Definition Classes
AnyRef → Any
def unit: Matrix

Returns the identity matrix of this matrix space, if one exists.
final def wait(): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
final def wait(arg0: Long, arg1: Int): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
final def wait(arg0: Long): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
def zero: Matrix

Returns the additive identity of this matrix space.
Returns the additive identity of this matrix space.

Definition Classes
FMxN → VectorSpace

trait FMxN extends VectorSpace

Type Members

type Col = (Col)#Vector

abstract type Matrix <: MatrixFMxN

trait MatrixFMxN extends VectorElement

type Row = (Row)#Vector

type Scalar = Element

type Transpose = (Transpose)#Matrix

type Vector = Matrix

trait VectorElement extends Any

Abstract Value Members

abstract val Col: FN { val Scalar: FMxN.this.Scalar.type }

abstract val Row: FN { val Scalar: FMxN.this.Scalar.type }

abstract val Scalar: Ring

implicit abstract def ScalarTag: ClassTag[Scalar]

abstract val Transpose: FMxN { ... /* 3 definitions in type refinement */ }

abstract def apply(entries: Array[Scalar]): Matrix

Concrete Value Members

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: Any): Boolean

final def asInstanceOf[T0]: T0

def clone(): AnyRef

def cols(cols: Col*): Matrix

def dim: Int

final def eq(arg0: AnyRef): Boolean

def equals(arg0: Any): Boolean

def finalize(): Unit

final def getClass(): Class[_]

def hashCode(): Int

final def isInstanceOf[T0]: Boolean

final def ne(arg0: AnyRef): Boolean

final def notify(): Unit

final def notifyAll(): Unit

def product[A <: FMxN { val Scalar: FMxN.this.Scalar.type }, B <: FMxN { val Scalar: FMxN.this.Scalar.type }](matrixA: product.A.MatrixFMxN, matrixB: product.B.MatrixFMxN): Matrix

def rows(rows: Row*): Matrix

final def synchronized[T0](arg0: ⇒ T0): T0

def toString(): String

def unit: Matrix

final def wait(): Unit

final def wait(arg0: Long, arg1: Int): Unit

final def wait(arg0: Long): Unit

def zero: Matrix

Inherited from VectorSpace

Inherited from AnyRef

Inherited from Any

Ungrouped