Matrix

The Matrix class provides support for matrices. A matrix can be initialized with a given size,

var a = Matrix(nrows,ncols)

where all elements are initially set to zero. Alternatively, a matrix can be created from an array,

var a = Matrix([[1,2], [3,4]])

or a Sparse matrix,

var a = Sparse([[0,0,1],[1,1,1],[2,2,1]])
var b = Matrix(a)

You can create a column vector like this,

var v = Matrix([1,2])

Finally, you can create a Matrix by assembling other matrices like this,

var a = Matrix([[0,1],[1,0]])
var b = Matrix([[a,0],[0,a]]) // produces a 4x4 matrix

Once a matrix is created, you can use all the regular arithmetic operators with matrix operands, e.g.

a+b
a*b

You can retrieved individual matrix entries with specified indices:

print a[0,0]

or create a submatrix using slices:

print a[0..1,0..1]

The division operator is used to solve a linear system, e.g.

var a = Matrix([[1,2],[3,4]])
var b = Matrix([1,2])

print b/a

yields the solution to the system a*x = b.

Assign

Copies the contents of matrix B into matrix A:

A.assign(B)

The two matrices must have the same dimensions.

Dimensions

Returns the dimensions of a matrix:

var A = Matrix([1,2,3]) // Create a column matrix 
print A.dimensions()    // Expect: [ 3, 1 ]

Eigenvalues

Returns a list of eigenvalues of a Matrix:

var A = Matrix([[0,1],[1,0]])
print A.eigenvalues() // Expect: [1,-1]

Eigensystem

Returns the eigenvalues and eigenvectors of a Matrix:

var A = Matrix([[0,1],[1,0]])
print A.eigensystem()

Eigensystem returns a two element list: The first element is a List of eigenvalues. The second element is a Matrix containing the corresponding eigenvectors as its columns:

print A.eigensystem()[0]
// [ 1, -1 ]
print A.eigensystem()[1]
// [ 0.707107 -0.707107 ]
// [ 0.707107 0.707107 ]

Inner

Computes the Frobenius inner product between two matrices:

var prod = A.inner(B)

Outer

Computes the outer produce between two vectors:

var prod = A.outer(B)

Note that outer always treats both vectors as column vectors.

Inverse

Returns the inverse of a matrix if it is invertible. Raises a MtrxSnglr error if the matrix is singular. E.g.

var m = Matrix([[1,2],[3,4]])
var mi = m.inverse()

yields the inverse of the matrix m, such that mi*m is the identity matrix.

Norm

Returns a matrix norm. By default the L2 norm is returned:

var a = Matrix([1,2,3,4])
print a.norm() // Expect: sqrt(30) = 5.47723...

You can select a different norm by supplying an argument:

import constants
print a.norm(1) // Expect: 10 (L1 norm is sum of absolute values) 
print a.norm(3) // Expect: 4.64159 (An unusual choice of norm)
print a.norm(Inf) // Expect: 4 (Inf-norm corresponds to maximum absolute value)

Reshape

Changes the dimensions of a matrix such that the total number of elements remains constant:

var A = Matrix([[1,3],[2,4]])
A.reshape(1,4) // 1 row, 4 columns
print A // Expect: [ 1, 2, 3, 4 ]

Note that elements are stored in column major-order.

Sum

Returns the sum of all entries in a matrix:

var sum = A.sum()

Transpose

Returns the transpose of a matrix:

var At = A.transpose()

Trace

Computes the trace (the sum of the diagonal elements) of a square matrix:

var tr = A.trace()

Roll

Rotates values in a Matrix about a given axis by a given shift:

var r = A.roll(shift, axis)

Elements that roll beyond the last position are re-introduced at the first.

IdentityMatrix

Constructs an identity matrix of a specified size:

var a = IdentityMatrix(size)

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