JScience v4.3

## org.jscience.mathematics.vector Class Matrix<F extends Field<F>>

```java.lang.Object org.jscience.mathematics.vector.Matrix<F>
```
All Implemented Interfaces:
javolution.lang.Immutable, javolution.lang.Realtime, javolution.lang.ValueType, GroupAdditive<Matrix<F>>, Ring<Matrix<F>>, Structure<Matrix<F>>, VectorSpace<Matrix<F>,F>
Direct Known Subclasses:
ComplexMatrix, DenseMatrix, Float64Matrix, SparseMatrix

`public abstract class Matrix<F extends Field<F>>extends java.lang.Objectimplements VectorSpace<Matrix<F>,F>, Ring<Matrix<F>>, javolution.lang.ValueType, javolution.lang.Realtime`

This class represents a rectangular table of elements of a ring-like algebraic structure.

Instances of this class can be used to resolve system of linear equations involving any kind of `Field` elements (e.g. `Real`, `Complex`, `Amount<?>`, `Function`, etc). For example:

``````        // Creates a dense matrix (2x2) of Rational numbers.
DenseMatrix<Rational> M = DenseMatrix.valueOf(
{ Rational.valueOf(23, 45), Rational.valueOf(33, 75) },
{ Rational.valueOf(15, 31), Rational.valueOf(-20, 45)});

// Creates a sparse matrix (16x2) of Real numbers.
SparseMatrix<Real> M = SparseMatrix.valueOf(
SparseVector.valueOf(16, Real.ZERO, 0, Real.valueOf(5)),
SparseVector.valueOf(16, Real.ZERO, 15, Real.valueOf(-3)));

// Creates a floating-point (64 bits) matrix (3x2).
Float64Matrix M = Float64Matrix.valueOf(
{{ 1.0, 2.0, 3.0}, { 4.0, 5.0, 6.0}});

// Creates a complex single column matrix (1x2).
ComplexMatrix M = ComplexMatrix.valueOf(
{{ Complex.valueOf(1.0, 2.0), Complex.valueOf(4.0, 5.0)}}).transpose();

// Creates an identity matrix (2x2) for modulo integer.
SparseMatrix<ModuloInteger> IDENTITY = SparseMatrix.valueOf(
DenseVector.valueOf(ModuloInteger.ONE, ModuloInteger.ONE), ModuloInteger.ZERO);
``````

Non-commutative field multiplication is supported. Invertible square matrices may form a non-commutative field (also called a division ring). In which case this class may be used to resolve system of linear equations with matrix coefficients.

Implementation Note: Matrices may use `StackContext` and `ConcurrentContext` in order to minimize heap allocation and accelerate calculations on multi-core systems.

Version:
3.3, December 24, 2006
Author:
Jean-Marie Dautelle
Wikipedia: Matrix (mathematics)

Field Summary
`protected static javolution.xml.XMLFormat<Matrix>` `XML`
Holds the default XML representation for matrices.

Constructor Summary
`protected ` `Matrix()`
Default constructor (for sub-classes).

Method Summary
`abstract  Matrix<F>` `adjoint()`
Returns the adjoint of this matrix.
`abstract  F` ```cofactor(int i, int j)```
Returns the cofactor of an element in this matrix.
`abstract  Matrix<F>` `copy()`
Returns a copy of this matrix `allocated` by the calling thread (possibly on the stack).
`abstract  F` `determinant()`
Returns the determinant of this matrix.
` Matrix<F>` `divide(Matrix<F> that)`
Returns this matrix divided by the one specified.
` boolean` ```equals(Matrix<F> that, java.util.Comparator<F> cmp)```
Indicates if this matrix can be considered equals to the one specified using the specified comparator when testing for element equality.
` boolean` `equals(java.lang.Object that)`
Indicates if this matrix is strictly equal to the object specified.
`abstract  F` ```get(int i, int j)```
Returns a single element from this matrix.
`abstract  Vector<F>` `getColumn(int j)`
Returns the column identified by the specified index in this matrix.
`abstract  Vector<F>` `getDiagonal()`
Returns the diagonal vector.
`abstract  int` `getNumberOfColumns()`
Returns the number of columns `n` for this matrix.
`abstract  int` `getNumberOfRows()`
Returns the number of rows `m` for this matrix.
`abstract  Vector<F>` `getRow(int i)`
Returns the row identified by the specified index in this matrix.
` int` `hashCode()`
Returns a hash code value for this matrix.
`abstract  Matrix<F>` `inverse()`
Returns the inverse of this matrix (must be square).
` boolean` `isSquare()`
Indicates if this matrix is square.
` Matrix<F>` `minus(Matrix<F> that)`
Returns the difference between this matrix and the one specified.
`abstract  Matrix<F>` `opposite()`
Returns the negation of this matrix.
`abstract  Matrix<F>` `plus(Matrix<F> that)`
Returns the sum of this matrix with the one specified.
` Matrix<F>` `pow(int exp)`
Returns this matrix raised at the specified exponent.
` Matrix<F>` `pseudoInverse()`
Returns the inverse or pseudo-inverse if this matrix if not square.
` Matrix<F>` `solve(Matrix<F> y)`
Solves this matrix for the specified matrix (returns `x` such as `this · x = y`).
` Vector<F>` `solve(Vector<F> y)`
Solves this matrix for the specified vector (returns `x` such as `this · x = y`).
`abstract  Matrix<F>` `tensor(Matrix<F> that)`
Returns the linear algebraic matrix tensor product of this matrix and another (Kronecker product).
`abstract  Matrix<F>` `times(F k)`
Returns the product of this matrix by the specified factor.
`abstract  Matrix<F>` `times(Matrix<F> that)`
Returns the product of this matrix with the one specified.
`abstract  Vector<F>` `times(Vector<F> v)`
Returns the product of this matrix by the specified vector.
` java.lang.String` `toString()`
Returns the text representation of this matrix as a `java.lang.String`.
` javolution.text.Text` `toText()`
Returns the text representation of this matrix.
` F` `trace()`
Returns the trace of this matrix.
`abstract  Matrix<F>` `transpose()`
Returns the transpose of this matrix.
`abstract  Vector<F>` `vectorization()`
Returns the vectorization of this matrix.

Methods inherited from class java.lang.Object
`clone, finalize, getClass, notify, notifyAll, wait, wait, wait`

Field Detail

### XML

`protected static final javolution.xml.XMLFormat<Matrix> XML`
Holds the default XML representation for matrices. For example:``` <DenseMatrix rows="2" columns="2"> <Complex real="1.0" imaginary="0.0" /> <Complex real="0.0" imaginary="1.0" /> <Complex real="0.0" imaginary="0.4" /> <Complex real="-5.0" imaginary="-1.0" /> </DenseMatrix>```

Constructor Detail

### Matrix

`protected Matrix()`
Default constructor (for sub-classes).

Method Detail

### getNumberOfRows

`public abstract int getNumberOfRows()`
Returns the number of rows `m` for this matrix.

Returns:
m, the number of rows.

### getNumberOfColumns

`public abstract int getNumberOfColumns()`
Returns the number of columns `n` for this matrix.

Returns:
n, the number of columns.

### get

```public abstract F get(int i,
int j)```
Returns a single element from this matrix.

Parameters:
`i` - the row index (range [0..m[).
`j` - the column index (range [0..n[).
Returns:
Throws:
`java.lang.IndexOutOfBoundsException` - ``` ((i < 0) || (i >= m)) || ((j < 0) || (j >= n))```

### getRow

`public abstract Vector<F> getRow(int i)`
Returns the row identified by the specified index in this matrix.

Parameters:
`i` - the row index (range [0..m[).
Returns:
the vector holding the specified row.
Throws:
`java.lang.IndexOutOfBoundsException` - `(i < 0) || (i >= m)`

### getColumn

`public abstract Vector<F> getColumn(int j)`
Returns the column identified by the specified index in this matrix.

Parameters:
`j` - the column index (range [0..n[).
Returns:
the vector holding the specified column.
Throws:
`java.lang.IndexOutOfBoundsException` - `(j < 0) || (j >= n)`

### getDiagonal

`public abstract Vector<F> getDiagonal()`
Returns the diagonal vector.

Returns:
the vector holding the diagonal elements.

### opposite

`public abstract Matrix<F> opposite()`
Returns the negation of this matrix.

Specified by:
`opposite` in interface `GroupAdditive<Matrix<F extends Field<F>>>`
Returns:
`-this`.

### plus

`public abstract Matrix<F> plus(Matrix<F> that)`
Returns the sum of this matrix with the one specified.

Specified by:
`plus` in interface `GroupAdditive<Matrix<F extends Field<F>>>`
Parameters:
`that` - the matrix to be added.
Returns:
`this + that`.
Throws:
`DimensionException` - matrices's dimensions are different.

### minus

`public Matrix<F> minus(Matrix<F> that)`
Returns the difference between this matrix and the one specified.

Parameters:
`that` - the matrix to be subtracted.
Returns:
`this - that`.
Throws:
`DimensionException` - matrices's dimensions are different.

### times

`public abstract Matrix<F> times(F k)`
Returns the product of this matrix by the specified factor.

Specified by:
`times` in interface `VectorSpace<Matrix<F extends Field<F>>,F extends Field<F>>`
Parameters:
`k` - the coefficient multiplier.
Returns:
`this · k`

### times

`public abstract Vector<F> times(Vector<F> v)`
Returns the product of this matrix by the specified vector.

Parameters:
`v` - the vector.
Returns:
`this · v`
Throws:
`DimensionException` - if ``` v.getDimension() != this.getNumberOfColumns()```
``` ```
``` ```
``` times public abstract Matrix<F> times(Matrix<F> that) Returns the product of this matrix with the one specified. Specified by:times in interface Ring<Matrix<F extends Field<F>>> Parameters:that - the matrix multiplier. Returns:this · that. Throws: DimensionException - if this.getNumberOfColumns() != that.getNumberOfRows(). inverse public abstract Matrix<F> inverse() Returns the inverse of this matrix (must be square). Returns:1 / this Throws: DimensionException - if this matrix is not square. divide public Matrix<F> divide(Matrix<F> that) Returns this matrix divided by the one specified. Parameters:that - the matrix divisor. Returns:this / that. Throws: DimensionException - if that matrix is not square or dimensions do not match. pseudoInverse public Matrix<F> pseudoInverse() Returns the inverse or pseudo-inverse if this matrix if not square. Note: To resolve the equation A * X = B, it is usually faster to calculate A.lu().solve(B) rather than A.inverse().times(B). Returns:the inverse or pseudo-inverse of this matrix. determinant public abstract F determinant() Returns the determinant of this matrix. Returns:this matrix determinant. Throws: DimensionException - if this matrix is not square. transpose public abstract Matrix<F> transpose() Returns the transpose of this matrix. Returns:A'. cofactor public abstract F cofactor(int i, int j) Returns the cofactor of an element in this matrix. It is the value obtained by evaluating the determinant formed by the elements not in that particular row or column. Parameters:i - the row index.j - the column index. Returns:the cofactor of THIS[i,j]. Throws: DimensionException - matrix is not square or its dimension is less than 2. adjoint public abstract Matrix<F> adjoint() Returns the adjoint of this matrix. It is obtained by replacing each element in this matrix with its cofactor and applying a + or - sign according (-1)**(i+j), and then finding the transpose of the resulting matrix. Returns:the adjoint of this matrix. Throws: DimensionException - if this matrix is not square or if its dimension is less than 2. isSquare public boolean isSquare() Indicates if this matrix is square. Returns:getNumberOfRows() == getNumberOfColumns() solve public Vector<F> solve(Vector<F> y) Solves this matrix for the specified vector (returns x such as this · x = y). Parameters:y - the vector for which the solution is calculated. Returns:x such as this · x = y Throws: DimensionException - if that matrix is not square or dimensions do not match. solve public Matrix<F> solve(Matrix<F> y) Solves this matrix for the specified matrix (returns x such as this · x = y). Parameters:y - the matrix for which the solution is calculated. Returns:x such as this · x = y Throws: DimensionException - if that matrix is not square or dimensions do not match. pow public Matrix<F> pow(int exp) Returns this matrix raised at the specified exponent. Parameters:exp - the exponent. Returns:thisexp Throws: DimensionException - if this matrix is not square. trace public F trace() Returns the trace of this matrix. Returns:the sum of the diagonal elements. tensor public abstract Matrix<F> tensor(Matrix<F> that) Returns the linear algebraic matrix tensor product of this matrix and another (Kronecker product). The default implementation returns a DenseMatrix. Parameters:that - the second matrix. Returns:this ⊗ thatSee Also: Wikipedia: Kronecker Product vectorization public abstract Vector<F> vectorization() Returns the vectorization of this matrix. The vectorization of a matrix is the column vector obtain by stacking the columns of the matrix on top of one another. The default implementation returns a DenseVector. Returns:the vectorization of this matrix.See Also: Wikipedia: Vectorization. toText public javolution.text.Text toText() Returns the text representation of this matrix. Specified by:toText in interface javolution.lang.Realtime Returns:the text representation of this matrix. toString public final java.lang.String toString() Returns the text representation of this matrix as a java.lang.String. Overrides:toString in class java.lang.Object Returns:toText().toString() equals public boolean equals(Matrix<F> that, java.util.Comparator<F> cmp) Indicates if this matrix can be considered equals to the one specified using the specified comparator when testing for element equality. The specified comparator may allow for some tolerance in the difference between the matrix elements. Parameters:that - the matrix to compare for equality.cmp - the comparator to use when testing for element equality. Returns:true if this matrix and the specified matrix are both matrices with equal elements according to the specified comparator; false otherwise. equals public boolean equals(java.lang.Object that) Indicates if this matrix is strictly equal to the object specified. Overrides:equals in class java.lang.Object Parameters:that - the object to compare for equality. Returns:true if this matrix and the specified object are both matrices with equal elements; false otherwise.See Also:equals(Matrix, Comparator) hashCode public int hashCode() Returns a hash code value for this matrix. Equals objects have equal hash codes. Overrides:hashCode in class java.lang.Object Returns:this matrix hash code value.See Also:equals(org.jscience.mathematics.vector.Matrix, java.util.Comparator) copy public abstract Matrix<F> copy() Returns a copy of this matrix allocated by the calling thread (possibly on the stack). Specified by:copy in interface javolution.lang.ValueType Returns:an identical and independant copy of this matrix. 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