Matrix (ELKI: Environment for DeveLoping KDD-Applications Supported by Index-Structures)

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de.lmu.ifi.dbs.elki.math.linearalgebra
Class Matrix

java.lang.Object
  de.lmu.ifi.dbs.elki.logging.AbstractLoggable
      de.lmu.ifi.dbs.elki.math.linearalgebra.Matrix

All Implemented Interfaces:: Loggable, Serializable, Cloneable

Direct Known Subclasses:: Vector

public class Matrix
extends AbstractLoggable
implements Cloneable, Serializable
extends AbstractLoggable
implements Cloneable, Serializable

The Matrix Class represents real-valued matrices.

For a Matrix M we have therefore M ∈$real;<sup>m × n</sup>, where m and n are the number of rows and columns, respectively.

See Also:: Serialized Form

Field Summary
`private int`	`columndimension` Column dimension.
`static double`	`DELTA` A small number to handle numbers near 0 as 0.
`private double[][]`	`elements` Array for internal storage of elements.
`private int`	`rowdimension` Row dimension.

Fields inherited from class de.lmu.ifi.dbs.elki.logging.AbstractLoggable
`debug`

Constructor Summary
`private`	`Matrix()` Basic-constructor for use in complex constructors only.
	`Matrix(double[][] elements)` Constructs a matrix from a 2-D array.
	`Matrix(double[] values, int m)` Construct a matrix from a one-dimensional packed array
	`Matrix(int m, int n)` Constructs an m-by-n matrix of zeros.
	`Matrix(int m, int n, double s)` Constructs an m-by-n constant matrix.
	`Matrix(RationalNumber[][] q)` Constructs a Matrix for a given array of arrays of `RationalNumber`s.

Method Summary
`double`	`angle(int colA, Matrix B, int colB)` Returns the angle of the colA col of this and the colB col of B.
`Matrix`	`appendColumns(Matrix columns)` Returns a matrix which consists of this matrix and the specified columns.
`Matrix`	`arrayLeftDivide(Matrix B)` Element-by-element left division, C = A.
`Matrix`	`arrayLeftDivideEquals(Matrix B)` Element-by-element left division in place, A = A.
`Matrix`	`arrayRightDivide(Matrix B)` Element-by-element right division, C = A.
`Matrix`	`arrayRightDivideEquals(Matrix B)` Element-by-element right division in place, A = A.
`Matrix`	`arrayTimes(Matrix B)` Element-by-element multiplication, C = A.
`Matrix`	`arrayTimesEquals(Matrix B)` Element-by-element multiplication in place, A = A.
`Matrix`	`cheatToAvoidSingularity(double constant)` Adds a given value to the diagonal entries if the entry is smaller than the constant.
`private void`	`checkMatrixDimensions(Matrix B)` Check if size(A) == size(B) *
`CholeskyDecomposition`	`chol()` Cholesky Decomposition
`Object`	`clone()` Clone the Matrix object.
`Matrix`	`completeBasis()` Completes this d x c basis of a subspace of R^d to a d x d basis of R^d, i.e. appends c-d columns to this basis.
`Matrix`	`completeToOrthonormalBasis()` Completes this d x c basis of a subspace of R^d to a d x d basis of R^d, i.e. appends c-d columns to this basis.
`double`	`cond()` Matrix condition (2 norm)
`static Matrix`	`constructWithCopy(double[][] A)` Construct a matrix from a copy of a 2-D array.
`Matrix`	`copy()` Make a deep copy of a matrix.
`double`	`det()` Matrix determinant
`static Matrix`	`diagonal(double[] diagonal)` Returns a quadratic Matrix consisting of zeros and of the given values on the diagonal.
`static Matrix`	`diagonal(Vector diagonal)` Returns a quadratic Matrix consisting of zeros and of the given values on the diagonal.
`String`	`dimensionInfo()` Returns the dimensionality of this matrix as a string.
`double`	`distanceCov(Matrix B)` distanceCov returns distance of two Matrices A and B, i.e. the root of the sum of the squared distances A_ij-B_ij.
`EigenvalueDecomposition`	`eig()` Eigenvalue Decomposition
`boolean`	`equals(Object obj)`
`double`	`euclideanNorm(int col)` Returns the euclidean norm of the col column.
`private RationalNumber[][]`	`exactGaussElimination()` Performes an exact Gauss-elimination of this Matrix using RationalNumbers to yield highest possible accuracy.
`private static RationalNumber[][]`	`exactGaussElimination(RationalNumber[][] gauss)` Performes recursivly Gauss-elimination on the given matrix of RationalNumbers.
`Matrix`	`exactGaussJordanElimination()` Returns a matrix derived by Gauss-Jordan-elimination using RationalNumbers for the transformations.
`private Matrix`	`gaussElimination()` Recursive gauss-elimination (non-reduced form).
`Matrix`	`gaussJordanElimination()` Deprecated. use LinearEquationSystem instead
`double`	`get(int i, int j)` Get a single element.
`double[][]`	`getArray()` Access the internal two-dimensional array.
`double[][]`	`getArrayCopy()` Copy the internal two-dimensional array.
`Matrix`	`getColumn(int j)` Returns the `j`th column of this matrix.
`int`	`getColumnDimensionality()` Returns the dimensionality of the columns of this matrix.
`double[]`	`getColumnPackedCopy()` Make a one-dimensional column packed copy of the internal array.
`Vector`	`getColumnVector(int j)` Returns the `j`th column of this matrix as vector.
`double[]`	`getDiagonal()` getDiagonal returns array of diagonal-elements.
`Matrix`	`getMatrix(int[] r, int[] c)` Get a submatrix.
`Matrix`	`getMatrix(int[] r, int j0, int j1)` Get a submatrix.
`Matrix`	`getMatrix(int i0, int i1, int[] c)` Get a submatrix.
`Matrix`	`getMatrix(int i0, int i1, int j0, int j1)` Get a submatrix.
`Matrix`	`getRow(int i)` Returns the `i`th row of this matrix.
`int`	`getRowDimensionality()` Returns the dimensionality of the rows of this matrix.
`double[]`	`getRowPackedCopy()` Make a one-dimensional row packed copy of the internal array.
`Vector`	`getRowVector(int i)` Returns the `i`th row of this matrix as vector.
`int`	`hashCode()`
`static Matrix`	`identity(int m, int n)` Generate identity matrix
`void`	`increment(int i, int j, double s)` Increments a single element.
`Matrix`	`inverse()` Matrix inverse or pseudoinverse
`boolean`	`isSymmetric()` Returns true, if this matrix is symmetric, false otherwise.
`boolean`	`linearlyIndependent(Matrix columnMatrix)` Returns true if the specified column matrix `a` is linearly independent to the columns of this matrix.
`LUDecomposition`	`lu()` LU Decomposition
`Matrix`	`minus(Matrix B)` C = A - B
`Matrix`	`minusEquals(Matrix B)` A = A - B
`double`	`norm1()` One norm
`double`	`norm2()` Two norm
`void`	`normalizeColumns()` Normalizes the columns of this matrix to length of 1.0.
`double`	`normF()` Frobenius norm
`double`	`normInf()` Infinity norm
`Matrix`	`orthonormalize()` Returns an orthonormalization of this matrix.
`Matrix`	`plus(Matrix B)` C = A + B
`Matrix`	`plusEquals(Matrix B)` A = A + B
`void`	`print(int w, int d)` Print the matrix to stdout.
`void`	`print(NumberFormat format, int width)` Print the matrix to stdout.
`void`	`print(PrintWriter output, int w, int d)` Print the matrix to the output stream.
`void`	`print(PrintWriter output, NumberFormat format, int width)` Print the matrix to the output stream.
`Matrix`	`projection(Matrix v)` Projects this row vector into the subspace formed by the specified matrix v.
`QRDecomposition`	`qr()` QR Decomposition
`static Matrix`	`random(int m, int n)` Generate matrix with random elements
`int`	`rank()` Matrix rank
`static Matrix`	`read(BufferedReader input)` Read a matrix from a stream.
`double`	`scalarProduct(int colA, Matrix B, int colB)` Returns the scalar product of the colA cols of this and the colB col of B.
`void`	`scaleColumn(int j, double scale)` Scales the specified column with the specified factor.
`void`	`scaleColumns(double scale)` Scales the columns of this matrix with the specified factor.
`void`	`set(int i, int j, double s)` Set a single element.
`void`	`setColumn(int j, Matrix column)` Sets the `j`th column of this matrix to the specified column.
`void`	`setMatrix(int[] r, int[] c, Matrix X)` Set a submatrix.
`void`	`setMatrix(int[] r, int j0, int j1, Matrix X)` Set a submatrix.
`void`	`setMatrix(int i0, int i1, int[] c, Matrix X)` Set a submatrix.
`void`	`setMatrix(int i0, int i1, int j0, int j1, Matrix X)` Set a submatrix.
`Matrix`	`solve(Matrix B)` Solve A*X = B
`Matrix`	`solveTranspose(Matrix B)` Solve XA = B, which is also A'X' = B'
`SingularValueDecomposition`	`svd()` Singular Value Decomposition
`void`	`swapColumn(int i, int j)` Swaps the specified columns of this matrix.
`void`	`swapRow(int i, int j)` Swaps the specified rows of this matrix.
`Matrix`	`times(double s)` Multiply a matrix by a scalar, C = s*A
`Matrix`	`times(Matrix B)` Linear algebraic matrix multiplication, A * B
`Matrix`	`timesEquals(double s)` Multiply a matrix by a scalar in place, A = s*A
`String`	`toString()` toString returns String-representation of Matrix.
`String`	`toString(int w, int d)` Returns a string representation of this matrix.
`String`	`toString(NumberFormat nf)` returns String-representation of Matrix.
`String`	`toString(String pre)` Returns a string representation of this matrix.
`String`	`toString(String pre, NumberFormat nf)` Returns a string representation of this matrix.
`double`	`trace()` Matrix trace.
`Matrix`	`transpose()` Matrix transpose.
`Matrix`	`uminus()` Unary minus
`static Matrix`	`unitMatrix(int dim)` Returns the unit matrix of the specified dimension.
`static Matrix`	`zeroMatrix(int dim)` Returns the zero matrix of the specified dimension.

Methods inherited from class de.lmu.ifi.dbs.elki.logging.AbstractLoggable
`debugFine, debugFiner, debugFinest, exception, message, progress, progress, progress, verbose, verbose, warning`

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

Field Detail

DELTA

public static final double DELTA

A small number to handle numbers near 0 as 0.

See Also:: Constant Field Values

elements

private double[][] elements

Array for internal storage of elements.

rowdimension

private int rowdimension

Row dimension.

columndimension

private int columndimension

Column dimension.

Constructor Detail

Matrix

private Matrix()

Basic-constructor for use in complex constructors only. Sets the debug-status for the class according to the present status of LoggingConfiguration.DEBUG

Matrix

public Matrix(int m,
              int n)

Constructs an m-by-n matrix of zeros.

Parameters:: m - number of rows; n - number of colums

Matrix

public Matrix(int m,
              int n,
              double s)

Constructs an m-by-n constant matrix.

Parameters:: m - number of rows; n - number of colums; s - A scalar value defining the constant value in the matrix

Matrix

public Matrix(double[][] elements)

Constructs a matrix from a 2-D array.

Parameters:: elements - an array of arrays of doubles defining the values of the matrix
Throws:: IllegalArgumentException - if not all rows conform in the same length

Matrix

public Matrix(RationalNumber[][] q)

Constructs a Matrix for a given array of arrays of RationalNumbers.

Parameters:: q - an array of arrays of RationalNumbers. q is not checked for consistency (i.e. whether all rows are of equal length)

Matrix

public Matrix(double[] values,
              int m)

Construct a matrix from a one-dimensional packed array

Parameters:: values - One-dimensional array of doubles, packed by columns (ala Fortran).; m - Number of rows.
Throws:: IllegalArgumentException - Array length must be a multiple of m.

Method Detail

constructWithCopy

public static Matrix constructWithCopy(double[][] A)

Construct a matrix from a copy of a 2-D array.

Parameters:: A - Two-dimensional array of doubles.
Throws:: IllegalArgumentException - All rows must have the same length

copy

public Matrix copy()

Make a deep copy of a matrix.

Returns:: a new matrix containing the same values as this matrix

clone

public Object clone()

Clone the Matrix object.

Overrides:: clone in class Object

getArray

public double[][] getArray()

Access the internal two-dimensional array.

Returns:: Pointer to the two-dimensional array of matrix elements.

getArrayCopy

public double[][] getArrayCopy()

Copy the internal two-dimensional array.

Returns:: Two-dimensional array copy of matrix elements.

getColumnPackedCopy

public double[] getColumnPackedCopy()

Make a one-dimensional column packed copy of the internal array.

Returns:: Matrix elements packed in a one-dimensional array by columns.

getRowPackedCopy

public double[] getRowPackedCopy()

Make a one-dimensional row packed copy of the internal array.

Returns:: Matrix elements packed in a one-dimensional array by rows.

getRowDimensionality

public int getRowDimensionality()

Returns the dimensionality of the rows of this matrix.

Returns:: m, the number of rows.

getColumnDimensionality

public int getColumnDimensionality()

Returns the dimensionality of the columns of this matrix.

Returns:: n, the number of columns.

get

public double get(int i,
                  int j)

Get a single element.

Parameters:: i - Row index.; j - Column index.
Returns:: A(i,j)
Throws:: ArrayIndexOutOfBoundsException

getMatrix

public Matrix getMatrix(int i0,
                        int i1,
                        int j0,
                        int j1)

Get a submatrix.

Parameters:: i0 - Initial row index; i1 - Final row index; j0 - Initial column index; j1 - Final column index
Returns:: A(i0:i1,j0:j1)
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

getMatrix

public Matrix getMatrix(int[] r,
                        int[] c)

Get a submatrix.

Parameters:: r - Array of row indices.; c - Array of column indices.
Returns:: A(r(:),c(:))
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

getMatrix

public Matrix getMatrix(int i0,
                        int i1,
                        int[] c)

Get a submatrix.

Parameters:: i0 - Initial row index; i1 - Final row index; c - Array of column indices.
Returns:: A(i0:i1,c(:))
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

getMatrix

public Matrix getMatrix(int[] r,
                        int j0,
                        int j1)

Get a submatrix.

Parameters:: r - Array of row indices.; j0 - Initial column index; j1 - Final column index
Returns:: A(r(:),j0:j1)
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

set

public void set(int i,
                int j,
                double s)

Set a single element.

Parameters:: i - Row index.; j - Column index.; s - A(i,j).
Throws:: ArrayIndexOutOfBoundsException

increment

public void increment(int i,
                      int j,
                      double s)

Increments a single element.

Parameters:: i - the row index; j - the column index; s - the increment value: A(i,j) = A(i.j) + s.
Throws:: ArrayIndexOutOfBoundsException

setMatrix

public void setMatrix(int i0,
                      int i1,
                      int j0,
                      int j1,
                      Matrix X)

Set a submatrix.

Parameters:: i0 - Initial row index; i1 - Final row index; j0 - Initial column index; j1 - Final column index; X - A(i0:i1,j0:j1)
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public void setMatrix(int[] r,
                      int[] c,
                      Matrix X)

Set a submatrix.

Parameters:: r - Array of row indices.; c - Array of column indices.; X - A(r(:),c(:))
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public void setMatrix(int[] r,
                      int j0,
                      int j1,
                      Matrix X)

Set a submatrix.

Parameters:: r - Array of row indices.; j0 - Initial column index; j1 - Final column index; X - A(r(:),j0:j1)
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public void setMatrix(int i0,
                      int i1,
                      int[] c,
                      Matrix X)

Set a submatrix.

Parameters:: i0 - Initial row index; i1 - Final row index; c - Array of column indices.; X - A(i0:i1,c(:))
Throws:: ArrayIndexOutOfBoundsException - Submatrix indices

getColumn

public Matrix getColumn(int j)

Returns the jth column of this matrix.

Parameters:: j - the index of the column to be returned
Returns:: the jth column of this matrix

getColumnVector

public Vector getColumnVector(int j)

Returns the jth column of this matrix as vector.

Parameters:: j - the index of the column to be returned
Returns:: the jth column of this matrix

getRow

public Matrix getRow(int i)

Returns the ith row of this matrix.

Parameters:: i - the index of the row to be returned
Returns:: the ith row of this matrix

getRowVector

public Vector getRowVector(int i)

Returns the ith row of this matrix as vector.

Parameters:: i - the index of the row to be returned
Returns:: the ith row of this matrix

setColumn

public void setColumn(int j,
                      Matrix column)

Sets the jth column of this matrix to the specified column.

Parameters:: j - the index of the column to be set; column - the value of the column to be set

transpose

public Matrix transpose()

Matrix transpose.

Returns:: A'

norm1

public double norm1()

One norm

Returns:: maximum column sum.

norm2

public double norm2()

Two norm

Returns:: maximum singular value.

normInf

public double normInf()

Infinity norm

Returns:: maximum row sum.

normF

public double normF()

Frobenius norm

Returns:: sqrt of sum of squares of all elements.

uminus

public Matrix uminus()

Unary minus

Returns:: -A

plus

public Matrix plus(Matrix B)

C = A + B

Parameters:: B - another matrix
Returns:: A + B

plusEquals

public Matrix plusEquals(Matrix B)

A = A + B

Parameters:: B - another matrix
Returns:: A + B

minus

public Matrix minus(Matrix B)

C = A - B

Parameters:: B - another matrix
Returns:: A - B

minusEquals

public Matrix minusEquals(Matrix B)

A = A - B

Parameters:: B - another matrix
Returns:: A - B

arrayTimes

public Matrix arrayTimes(Matrix B)

Element-by-element multiplication, C = A.*B

Parameters:: B - another matrix
Returns:: A.*B

arrayTimesEquals

public Matrix arrayTimesEquals(Matrix B)

Element-by-element multiplication in place, A = A.*B

Parameters:: B - another matrix
Returns:: A.*B

arrayRightDivide

public Matrix arrayRightDivide(Matrix B)

Element-by-element right division, C = A./B

Parameters:: B - another matrix
Returns:: A./B

arrayRightDivideEquals

public Matrix arrayRightDivideEquals(Matrix B)

Element-by-element right division in place, A = A./B

Parameters:: B - another matrix
Returns:: A./B

arrayLeftDivide

public Matrix arrayLeftDivide(Matrix B)

Element-by-element left division, C = A.\B

Parameters:: B - another matrix
Returns:: A.\B

arrayLeftDivideEquals

public Matrix arrayLeftDivideEquals(Matrix B)

Element-by-element left division in place, A = A.\B

Parameters:: B - another matrix
Returns:: A.\B

times

public Matrix times(double s)

Multiply a matrix by a scalar, C = s*A

Parameters:: s - scalar
Returns:: s*A

timesEquals

public Matrix timesEquals(double s)

Multiply a matrix by a scalar in place, A = s*A

Parameters:: s - scalar
Returns:: replace A by s*A

times

public Matrix times(Matrix B)

Linear algebraic matrix multiplication, A * B

Parameters:: B - another matrix
Returns:: Matrix product, A * B
Throws:: IllegalArgumentException - Matrix inner dimensions must agree.

lu

public LUDecomposition lu()

LU Decomposition

Returns:: LUDecomposition
See Also:: LUDecomposition

qr

public QRDecomposition qr()

QR Decomposition

Returns:: QRDecomposition
See Also:: QRDecomposition

chol

public CholeskyDecomposition chol()

Cholesky Decomposition

Returns:: CholeskyDecomposition
See Also:: CholeskyDecomposition

svd

public SingularValueDecomposition svd()

Singular Value Decomposition

Returns:: SingularValueDecomposition
See Also:: SingularValueDecomposition

eig

public EigenvalueDecomposition eig()

Eigenvalue Decomposition

Returns:: EigenvalueDecomposition
See Also:: EigenvalueDecomposition

solve

public Matrix solve(Matrix B)

Solve A*X = B

Parameters:: B - right hand side
Returns:: solution if A is square, least squares solution otherwise

solveTranspose

public Matrix solveTranspose(Matrix B)

Solve X*A = B, which is also A'*X' = B'

Parameters:: B - right hand side
Returns:: solution if A is square, least squares solution otherwise.

inverse

public Matrix inverse()

Matrix inverse or pseudoinverse

Returns:: inverse(A) if A is square, pseudoinverse otherwise.

det

public double det()

Matrix determinant

Returns:: determinant

rank

public int rank()

Matrix rank

Returns:: effective numerical rank, obtained from SVD.

cond

public double cond()

Matrix condition (2 norm)

Returns:: ratio of largest to smallest singular value.

trace

public double trace()

Matrix trace.

Returns:: sum of the diagonal elements.

random

public static Matrix random(int m,
                            int n)

Generate matrix with random elements

Parameters:: m - Number of rows.; n - Number of colums.
Returns:: An m-by-n matrix with uniformly distributed random elements.

identity

public static Matrix identity(int m,
                              int n)

Generate identity matrix

Parameters:: m - Number of rows.; n - Number of colums.
Returns:: An m-by-n matrix with ones on the diagonal and zeros elsewhere.

print

public void print(int w,
                  int d)

Print the matrix to stdout. Line the elements up in columns with a Fortran-like 'Fw.d' style format.

Parameters:: w - Column width.; d - Number of digits after the decimal.

print

public void print(PrintWriter output,
                  int w,
                  int d)

Print the matrix to the output stream. Line the elements up in columns with a Fortran-like 'Fw.d' style format.

Parameters:: output - Output stream.; w - Column width.; d - Number of digits after the decimal.

print

public void print(NumberFormat format,
                  int width)

Print the matrix to stdout. Line the elements up in columns. Use the format object, and right justify within columns of width characters. Note that is the matrix is to be read back in, you probably will want to use a NumberFormat that is set to US Locale.

Parameters:: format - A Formatting object for individual elements.; width - Field width for each column.
See Also:: DecimalFormat.setDecimalFormatSymbols(java.text.DecimalFormatSymbols)

print

public void print(PrintWriter output,
                  NumberFormat format,
                  int width)

Print the matrix to the output stream. Line the elements up in columns. Use the format object, and right justify within columns of width characters. Note that is the matrix is to be read back in, you probably will want to use a NumberFormat that is set to US Locale.

Parameters:: output - the output stream.; format - A formatting object to format the matrix elements; width - Column width.
See Also:: DecimalFormat.setDecimalFormatSymbols(java.text.DecimalFormatSymbols)

read

public static Matrix read(BufferedReader input)
                   throws IOException

Read a matrix from a stream. The format is the same the print method, so printed matrices can be read back in (provided they were printed using US Locale). Elements are separated by whitespace, all the elements for each row appear on a single line, the last row is followed by a blank line.

Parameters:: input - the input stream.
Throws:: IOException

checkMatrixDimensions

private void checkMatrixDimensions(Matrix B)

Check if size(A) == size(B) *

toString

public String toString(int w,
                       int d)

Returns a string representation of this matrix.

Parameters:: w - column width; d - number of digits after the decimal
Returns:: a string representation of this matrix

toString

public String toString()

toString returns String-representation of Matrix.

Overrides:: toString in class Object

toString

public String toString(String pre)

Returns a string representation of this matrix. In each line the specified String

pre<\code> is prefixed.



Parameters:
pre - the prefix of each line
Returns:
a string representation of this matrix





toString
public String toString(NumberFormat nf)

returns String-representation of Matrix.





Parameters:
nf - NumberFormat to specify output precision
Returns:
String representation of this Matrix in precision as specified by
         given NumberFormat





toString
public String toString(String pre,
                       NumberFormat nf)

Returns a string representation of this matrix. In each line the
 specified String pre<\code> is prefixed.





Parameters:
nf - number format for output accuracy
pre - the prefix of each line
Returns:
a string representation of this matrix





getDiagonal
public double[] getDiagonal()

getDiagonal returns array of diagonal-elements.






Returns:
double[] the values on the diagonaly of the Matrix





scaleColumns
public void scaleColumns(double scale)

Scales the columns of this matrix with the specified factor.





Parameters:
scale - the factor to scale the column





normalizeColumns
public void normalizeColumns()

Normalizes the columns of this matrix to length of 1.0.











scaleColumn
public void scaleColumn(int j,
                        double scale)

Scales the specified column with the specified factor.





Parameters:
j - the index of the column tp be scaled
scale - the factor to scale the column





distanceCov
public double distanceCov(Matrix B)

distanceCov returns distance of two Matrices A and B, i.e. the root of
 the sum of the squared distances A_ij-B_ij.





Parameters:
B - Matrix to compute distance from this (A)
Returns:
distance of Matrices





angle
public double angle(int colA,
                    Matrix B,
                    int colB)

Returns the angle of the colA col of this and the colB col of B.





Parameters:
colA - the column of A to compute angle for
B - second Matrix
colB - the column of B to compute angle for
Returns:
double angle of the first cols of this and B





scalarProduct
public double scalarProduct(int colA,
                            Matrix B,
                            int colB)

Returns the scalar product of the colA cols of this and the colB col of
 B.





Parameters:
colA - The column of A to compute scalar product for
B - second Matrix
colB - The column of B to compute scalar product for
Returns:
double The scalar product of the first cols of this and B





euclideanNorm
public double euclideanNorm(int col)

Returns the euclidean norm of the col column.





Parameters:
col - The column to compute euclidean norm of
Returns:
double





diagonal
public static Matrix diagonal(double[] diagonal)

Returns a quadratic Matrix consisting of zeros and of the given values on
 the diagonal.





Parameters:
diagonal - the values on the diagonal
Returns:
the resulting matrix





diagonal
public static Matrix diagonal(Vector diagonal)

Returns a quadratic Matrix consisting of zeros and of the given values on
 the diagonal.





Parameters:
diagonal - the values on the diagonal
Returns:
the resulting matrix





swapRow
public void swapRow(int i,
                    int j)

Swaps the specified rows of this matrix.





Parameters:
i - the first row
j - the second row





swapColumn
public void swapColumn(int i,
                       int j)

Swaps the specified columns of this matrix.





Parameters:
i - the first column
j - the second column





projection
public Matrix projection(Matrix v)

Projects this row vector into the subspace formed by the specified matrix
 v.





Parameters:
v - the subspace matrix
Returns:
the projection of p into the subspace formed by v
Throws:
IllegalArgumentException - if this matrix is no row vector, i.e.
                                  this matrix has more than one column or this matrix and v have
                                  different length of rows





unitMatrix
public static Matrix unitMatrix(int dim)

Returns the unit matrix of the specified dimension.





Parameters:
dim - the dimensionality of the unit matrix
Returns:
the unit matrix of the specified dimension





zeroMatrix
public static Matrix zeroMatrix(int dim)

Returns the zero matrix of the specified dimension.





Parameters:
dim - the dimensionality of the unit matrix
Returns:
the zero matrix of the specified dimension





linearlyIndependent
public boolean linearlyIndependent(Matrix columnMatrix)

Returns true if the specified column matrix a is linearly
 independent to the columns of this matrix. Linearly independence is
 given, if the matrix resulting from appending a to this
 matrix has full rank.





Parameters:
columnMatrix - the column matrix to be tested for linear
                     independence
Returns:
true if the specified column matrix is linearly independent to
         the columns of this matrix





exactGaussJordanElimination
public Matrix exactGaussJordanElimination()

Returns a matrix derived by Gauss-Jordan-elimination using
 RationalNumbers for the transformations.






Returns:
a matrix derived by Gauss-Jordan-elimination using
         RationalNumbers for the transformations





gaussJordanElimination
@Deprecated
public Matrix gaussJordanElimination()

Deprecated. use LinearEquationSystem instead

Returns a matrix derived by Gauss-Jordan-elimination.






Returns:
a new Matrix that is the result of Gauss-Jordan-elimination





exactGaussElimination
private RationalNumber[][] exactGaussElimination()

Performes an exact Gauss-elimination of this Matrix using RationalNumbers
 to yield highest possible accuracy.






Returns:
an array of arrays of RationalNumbers representing the
         Gauss-eliminated form of this Matrix





exactGaussElimination
private static RationalNumber[][] exactGaussElimination(RationalNumber[][] gauss)

Performes recursivly Gauss-elimination on the given matrix of
 RationalNumbers.





Parameters:
gauss - an array of arrays of RationalNumber
Returns:
recursivly derived Gauss-elimination-form of the given matrix of
         RationalNumbers





gaussElimination
private Matrix gaussElimination()

Recursive gauss-elimination (non-reduced form).






Returns:
a new Matrix that is the result of Gauss-elimination





isSymmetric
public boolean isSymmetric()

Returns true, if this matrix is symmetric, false otherwise.






Returns:
true, if this matrix is symmetric, false otherwise





dimensionInfo
public String dimensionInfo()

Returns the dimensionality of this matrix as a string.






Returns:
the dimensionality of this matrix as a string





completeBasis
public Matrix completeBasis()

Completes this d x c basis of a subspace of R^d to a d x d basis of R^d,
 i.e. appends c-d columns to this basis.






Returns:
the appended columns





completeToOrthonormalBasis
public Matrix completeToOrthonormalBasis()

Completes this d x c basis of a subspace of R^d to a d x d basis of R^d,
 i.e. appends c-d columns to this basis.






Returns:
the appended columns





appendColumns
public Matrix appendColumns(Matrix columns)

Returns a matrix which consists of this matrix and the specified columns.





Parameters:
columns - the columns to be appended
Returns:
the new matrix with the appended columns





orthonormalize
public Matrix orthonormalize()

Returns an orthonormalization of this matrix.






Returns:
the orthonormalized matrix





cheatToAvoidSingularity
public Matrix cheatToAvoidSingularity(double constant)

Adds a given value to the diagonal entries if the entry is smaller than
 the constant.





Parameters:
constant - value to add to the diagonal entries
Returns:
a new Matrix differing from this Matrix by the given value added
         to the diagonal entries





hashCode
public int hashCode()


Overrides:
hashCode in class Object








equals
public boolean equals(Object obj)


Overrides:
equals in class Object

















  
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                    Release
                   
                   
                    0.1
                   
				   
                    (2008-07-10_1838)
                   
                 
               
            





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de.lmu.ifi.dbs.elki.math.linearalgebra Class Matrix

DELTA

elements

rowdimension

columndimension

Matrix

Matrix

Matrix

Matrix

Matrix

Matrix

constructWithCopy

copy

clone

getArray

getArrayCopy

getColumnPackedCopy

getRowPackedCopy

getRowDimensionality

getColumnDimensionality

get

getMatrix

getMatrix

getMatrix

getMatrix

set

increment

setMatrix

setMatrix

setMatrix

setMatrix

getColumn

getColumnVector

getRow

getRowVector

setColumn

transpose

norm1

norm2

normInf

normF

uminus

plus

plusEquals

minus

minusEquals

arrayTimes

arrayTimesEquals

arrayRightDivide

arrayRightDivideEquals

arrayLeftDivide

arrayLeftDivideEquals

times

timesEquals

times

lu

qr

chol

svd

eig

solve

solveTranspose

inverse

det

rank

cond

trace

random

identity

print

print

print

print

read

checkMatrixDimensions

toString

toString

toString

toString

toString

de.lmu.ifi.dbs.elki.math.linearalgebra
Class Matrix