Linear Algebra

When SciPy is built using the optimized ATLAS LAPACK and BLAS libraries, it has very fast linear algebra capabilities. If you dig deep enough, all of the raw lapack and blas libraries are available for your use for even more speed. In this section, some easier-to-use interfaces to these routines are described.

All of these linear algebra routines expect an object that can be converted into a 2-dimensional array. The output of these routines is also a two-dimensional array. There is a matrix class defined in Numpy, which you can initialize with an appropriate Numpy array in order to get objects for which multiplication is matrix-multiplication instead of the default, element-by-element multiplication.

Matrix Class

The matrix class is initialized with the SciPy command mat which is just convenient short-hand for matrix. If you are going to be doing a lot of matrix-math, it is convenient to convert arrays into matrices using this command. One advantage of using the mat command is that you can enter two-dimensional matrices using MATLAB-like syntax with commas or spaces separating columns and semicolons separting rows as long as the matrix is placed in a string passed to mat .

Basic routines

Finding Inverse

The inverse of a matrix is the matrix such that where is the identity matrix consisting of ones down the main diagonal. Usually is denoted . In SciPy, the matrix inverse of the Numpy array, A, is obtained using linalg.inv (A) , or using A.I if A is a Matrix. For example, let

then

The following example demonstrates this computation in SciPy

>>> A = mat('[1 3 5; 2 5 1; 2 3 8]')
>>> A
matrix([[1, 3, 5],
        [2, 5, 1],
        [2, 3, 8]])
>>> A.I
matrix([[-1.48,  0.36,  0.88],
        [ 0.56,  0.08, -0.36],
        [ 0.16, -0.12,  0.04]])
>>> from scipy import linalg
>>> linalg.inv(A)
array([[-1.48,  0.36,  0.88],
       [ 0.56,  0.08, -0.36],
       [ 0.16, -0.12,  0.04]])

Solving linear system

Solving linear systems of equations is straightforward using the scipy command linalg.solve. This command expects an input matrix and a right-hand-side vector. The solution vector is then computed. An option for entering a symmetrix matrix is offered which can speed up the processing when applicable. As an example, suppose it is desired to solve the following simultaneous equations:

We could find the solution vector using a matrix inverse:

However, it is better to use the linalg.solve command which can be faster and more numerically stable. In this case it however gives the same answer as shown in the following example:

>>> A = mat('[1 3 5; 2 5 1; 2 3 8]')
>>> b = mat('[10;8;3]')
>>> A.I*b
matrix([[-9.28],
        [ 5.16],
        [ 0.76]])
>>> linalg.solve(A,b)
array([[-9.28],
       [ 5.16],
       [ 0.76]])

Finding Determinant

The determinant of a square matrix is often denoted and is a quantity often used in linear algebra. Suppose are the elements of the matrix and let be the determinant of the matrix left by removing the row and column from . Then for any row

This is a recursive way to define the determinant where the base case is defined by accepting that the determinant of a matrix is the only matrix element. In SciPy the determinant can be calculated with linalg.det . For example, the determinant of

is

In SciPy this is computed as shown in this example:

>>> A = mat('[1 3 5; 2 5 1; 2 3 8]')
>>> linalg.det(A)
-25.000000000000004

Computing norms

Matrix and vector norms can also be computed with SciPy. A wide range of norm definitions are available using different parameters to the order argument of linalg.norm . This function takes a rank-1 (vectors) or a rank-2 (matrices) array and an optional order argument (default is 2). Based on these inputs a vector or matrix norm of the requested order is computed.

For vector x , the order parameter can be any real number including inf or -inf. The computed norm is

For matrix the only valid values for norm are inf, and ‘fro’ (or ‘f’) Thus,

where are the singular values of .

Solving linear least-squares problems and pseudo-inverses

Linear least-squares problems occur in many branches of applied mathematics. In this problem a set of linear scaling coefficients is sought that allow a model to fit data. In particular it is assumed that data is related to data through a set of coefficients and model functions via the model

where represents uncertainty in the data. The strategy of least squares is to pick the coefficients to minimize

Theoretically, a global minimum will occur when

or

where

When is invertible, then

where is called the pseudo-inverse of Notice that using this definition of the model can be written

The command linalg.lstsq will solve the linear least squares problem for given and . In addition linalg.pinv or linalg.pinv2 (uses a different method based on singular value decomposition) will find given

The following example and figure demonstrate the use of linalg.lstsq and linalg.pinv for solving a data-fitting problem. The data shown below were generated using the model:

where for , , and Noise is added to and the coefficients and are estimated using linear least squares.

>>> from numpy import *
>>> from scipy import linalg
>>> import matplotlib.pyplot as plt

>>> c1,c2= 5.0,2.0
>>> i = r_[1:11]
>>> xi = 0.1*i
>>> yi = c1*exp(-xi)+c2*xi
>>> zi = yi + 0.05*max(yi)*random.randn(len(yi))

>>> A = c_[exp(-xi)[:,newaxis],xi[:,newaxis]]
>>> c,resid,rank,sigma = linalg.lstsq(A,zi)

>>> xi2 = r_[0.1:1.0:100j]
>>> yi2 = c[0]*exp(-xi2) + c[1]*xi2

>>> plt.plot(xi,zi,'x',xi2,yi2)
>>> plt.axis([0,1.1,3.0,5.5])
>>> plt.xlabel('$x_i$')
>>> plt.title('Data fitting with linalg.lstsq')
>>> plt.show()

(Source code)

Output

(PNG, PDF)

Generalized inverse

The generalized inverse is calculated using the command linalg.pinv or linalg.pinv2. These two commands differ in how they compute the generalized inverse. The first uses the linalg.lstsq algorithm while the second uses singular value decomposition. Let be an matrix, then if the generalized inverse is

while if matrix the generalized inverse is

In both cases for , then

as long as is invertible.

Decompositions

In many applications it is useful to decompose a matrix using other representations. There are several decompositions supported by SciPy.

Eigenvalues and eigenvectors

The eigenvalue-eigenvector problem is one of the most commonly employed linear algebra operations. In one popular form, the eigenvalue-eigenvector problem is to find for some square matrix scalars and corresponding vectors such that

For an matrix, there are (not necessarily distinct) eigenvalues — roots of the (characteristic) polynomial

The eigenvectors, , are also sometimes called right eigenvectors to distinguish them from another set of left eigenvectors that satisfy

or

With it’s default optional arguments, the command linalg.eig returns and However, it can also return and just by itself ( linalg.eigvals returns just as well).

In addtion, linalg.eig can also solve the more general eigenvalue problem

for square matrices and The standard eigenvalue problem is an example of the general eigenvalue problem for When a generalized eigenvalue problem can be solved, then it provides a decomposition of as

where is the collection of eigenvectors into columns and is a diagonal matrix of eigenvalues.

By definition, eigenvectors are only defined up to a constant scale factor. In SciPy, the scaling factor for the eigenvectors is chosen so that

As an example, consider finding the eigenvalues and eigenvectors of the matrix

The characteristic polynomial is

The roots of this polynomial are the eigenvalues of :

The eigenvectors corresponding to each eigenvalue can be found using the original equation. The eigenvectors associated with these eigenvalues can then be found.

>>> from scipy import linalg
>>> A = mat('[1 5 2; 2 4 1; 3 6 2]')
>>> la,v = linalg.eig(A)
>>> l1,l2,l3 = la
>>> print l1, l2, l3
(7.95791620491+0j) (-1.25766470568+0j) (0.299748500767+0j)

>>> print v[:,0]
[-0.5297175  -0.44941741 -0.71932146]
>>> print v[:,1]
[-0.90730751  0.28662547  0.30763439]
>>> print v[:,2]
[ 0.28380519 -0.39012063  0.87593408]
>>> print sum(abs(v**2),axis=0)
[ 1.  1.  1.]

>>> v1 = mat(v[:,0]).T
>>> print max(ravel(abs(A*v1-l1*v1)))
8.881784197e-16

Singular value decomposition

Singular Value Decompostion (SVD) can be thought of as an extension of the eigenvalue problem to matrices that are not square. Let be an matrix with and arbitrary. The matrices and are square hermitian matrices [1] of size and respectively. It is known that the eigenvalues of square hermitian matrices are real and non-negative. In addtion, there are at most identical non-zero eigenvalues of and Define these positive eigenvalues as The square-root of these are called singular values of The eigenvectors of are collected by columns into an unitary [2] matrix while the eigenvectors of are collected by columns in the unitary matrix , the singular values are collected in an zero matrix with main diagonal entries set to the singular values. Then

is the singular-value decomposition of Every matrix has a singular value decomposition. Sometimes, the singular values are called the spectrum of The command linalg.svd will return , , and as an array of the singular values. To obtain the matrix use linalg.diagsvd. The following example illustrates the use of linalg.svd .

>>> A = mat('[1 3 2; 1 2 3]')
>>> M,N = A.shape
>>> U,s,Vh = linalg.svd(A)
>>> Sig = mat(linalg.diagsvd(s,M,N))
>>> U, Vh = mat(U), mat(Vh)
>>> print U
[[-0.70710678 -0.70710678]
 [-0.70710678  0.70710678]]
>>> print Sig
[[ 5.19615242  0.          0.        ]
 [ 0.          1.          0.        ]]
>>> print Vh
[[ -2.72165527e-01  -6.80413817e-01  -6.80413817e-01]
 [ -6.18652536e-16  -7.07106781e-01   7.07106781e-01]
 [ -9.62250449e-01   1.92450090e-01   1.92450090e-01]]

>>> print A
[[1 3 2]
 [1 2 3]]
>>> print U*Sig*Vh
[[ 1.  3.  2.]
 [ 1.  2.  3.]]

[1]	A hermitian matrix satisfies

[2]	A unitary matrix satisfies so that

LU decomposition

The LU decompostion finds a representation for the matrix as

where is an permutation matrix (a permutation of the rows of the identity matrix), is in lower triangular or trapezoidal matrix ( ) with unit-diagonal, and is an upper triangular or trapezoidal matrix. The SciPy command for this decomposition is linalg.lu .

Such a decomposition is often useful for solving many simultaneous equations where the left-hand-side does not change but the right hand side does. For example, suppose we are going to solve

for many different . The LU decomposition allows this to be written as

Because is lower-triangular, the equation can be solved for and finally very rapidly using forward- and back-substitution. An initial time spent factoring allows for very rapid solution of similar systems of equations in the future. If the intent for performing LU decomposition is for solving linear systems then the command linalg.lu_factor should be used followed by repeated applications of the command linalg.lu_solve to solve the system for each new right-hand-side.

Cholesky decomposition

Cholesky decomposition is a special case of LU decomposition applicable to Hermitian positive definite matrices. When and for all , then decompositions of can be found so that

where is lower-triangular and is upper triangular. Notice that The command linagl.cholesky computes the cholesky factorization. For using cholesky factorization to solve systems of equations there are also linalg.cho_factor and linalg.cho_solve routines that work similarly to their LU decomposition counterparts.

QR decomposition

The QR decomposition (sometimes called a polar decomposition) works for any array and finds an unitary matrix and an upper-trapezoidal matrix such that

Notice that if the SVD of is known then the QR decomposition can be found

implies that and Note, however, that in SciPy independent algorithms are used to find QR and SVD decompositions. The command for QR decomposition is linalg.qr .

Schur decomposition

For a square matrix, , the Schur decomposition finds (not-necessarily unique) matrices and such that

where is a unitary matrix and is either upper-triangular or quasi-upper triangular depending on whether or not a real schur form or complex schur form is requested. For a real schur form both and are real-valued when is real-valued. When is a real-valued matrix the real schur form is only quasi-upper triangular because blocks extrude from the main diagonal corresponding to any complex- valued eigenvalues. The command linalg.schur finds the Schur decomposition while the command linalg.rsf2csf converts and from a real Schur form to a complex Schur form. The Schur form is especially useful in calculating functions of matrices.

The following example illustrates the schur decomposition:

>>> from scipy import linalg
>>> A = mat('[1 3 2; 1 4 5; 2 3 6]')
>>> T,Z = linalg.schur(A)
>>> T1,Z1 = linalg.schur(A,'complex')
>>> T2,Z2 = linalg.rsf2csf(T,Z)
>>> print T
[[ 9.90012467  1.78947961 -0.65498528]
 [ 0.          0.54993766 -1.57754789]
 [ 0.          0.51260928  0.54993766]]
>>> print T2
[[ 9.90012467 +0.00000000e+00j -0.32436598 +1.55463542e+00j
  -0.88619748 +5.69027615e-01j]
 [ 0.00000000 +0.00000000e+00j  0.54993766 +8.99258408e-01j
   1.06493862 +1.37016050e-17j]
 [ 0.00000000 +0.00000000e+00j  0.00000000 +0.00000000e+00j
   0.54993766 -8.99258408e-01j]]
>>> print abs(T1-T2) # different
[[  1.24357637e-14   2.09205364e+00   6.56028192e-01]
 [  0.00000000e+00   4.00296604e-16   1.83223097e+00]
 [  0.00000000e+00   0.00000000e+00   4.57756680e-16]]
>>> print abs(Z1-Z2) # different
[[ 0.06833781  1.10591375  0.23662249]
 [ 0.11857169  0.5585604   0.29617525]
 [ 0.12624999  0.75656818  0.22975038]]
>>> T,Z,T1,Z1,T2,Z2 = map(mat,(T,Z,T1,Z1,T2,Z2))
>>> print abs(A-Z*T*Z.H) # same
[[  1.11022302e-16   4.44089210e-16   4.44089210e-16]
 [  4.44089210e-16   1.33226763e-15   8.88178420e-16]
 [  8.88178420e-16   4.44089210e-16   2.66453526e-15]]
>>> print abs(A-Z1*T1*Z1.H) # same
[[  1.00043248e-15   2.22301403e-15   5.55749485e-15]
 [  2.88899660e-15   8.44927041e-15   9.77322008e-15]
 [  3.11291538e-15   1.15463228e-14   1.15464861e-14]]
>>> print abs(A-Z2*T2*Z2.H) # same
[[  3.34058710e-16   8.88611201e-16   4.18773089e-18]
 [  1.48694940e-16   8.95109973e-16   8.92966151e-16]
 [  1.33228956e-15   1.33582317e-15   3.55373104e-15]]

Matrix Functions

Consider the function with Taylor series expansion

A matrix function can be defined using this Taylor series for the square matrix as

While, this serves as a useful representation of a matrix function, it is rarely the best way to calculate a matrix function.

Exponential and logarithm functions

The matrix exponential is one of the more common matrix functions. It can be defined for square matrices as

The command linalg.expm3 uses this Taylor series definition to compute the matrix exponential. Due to poor convergence properties it is not often used.

Another method to compute the matrix exponential is to find an eigenvalue decomposition of :

and note that

where the matrix exponential of the diagonal matrix is just the exponential of its elements. This method is implemented in linalg.expm2 .

The preferred method for implementing the matrix exponential is to use scaling and a Padé approximation for . This algorithm is implemented as linalg.expm .

The inverse of the matrix exponential is the matrix logarithm defined as the inverse of the matrix exponential.

The matrix logarithm can be obtained with linalg.logm .

Trigonometric functions

The trigonometric functions , , and are implemented for matrices in linalg.sinm, linalg.cosm, and linalg.tanm respectively. The matrix sin and cosine can be defined using Euler’s identity as

The tangent is

and so the matrix tangent is defined as

Hyperbolic trigonometric functions

The hyperbolic trigonemetric functions , , and can also be defined for matrices using the familiar definitions:

These matrix functions can be found using linalg.sinhm, linalg.coshm , and linalg.tanhm.

Arbitrary function

Finally, any arbitrary function that takes one complex number and returns a complex number can be called as a matrix function using the command linalg.funm. This command takes the matrix and an arbitrary Python function. It then implements an algorithm from Golub and Van Loan’s book “Matrix Computations “to compute function applied to the matrix using a Schur decomposition. Note that the function needs to accept complex numbers as input in order to work with this algorithm. For example the following code computes the zeroth-order Bessel function applied to a matrix.

>>> from scipy import special, random, linalg
>>> A = random.rand(3,3)
>>> B = linalg.funm(A,lambda x: special.jv(0,x))
>>> print A
[[ 0.72578091  0.34105276  0.79570345]
 [ 0.65767207  0.73855618  0.541453  ]
 [ 0.78397086  0.68043507  0.4837898 ]]
>>> print B
[[ 0.72599893 -0.20545711 -0.22721101]
 [-0.27426769  0.77255139 -0.23422637]
 [-0.27612103 -0.21754832  0.7556849 ]]
>>> print linalg.eigvals(A)
[ 1.91262611+0.j  0.21846476+0.j -0.18296399+0.j]
>>> print special.jv(0, linalg.eigvals(A))
[ 0.27448286+0.j  0.98810383+0.j  0.99164854+0.j]
>>> print linalg.eigvals(B)
[ 0.27448286+0.j  0.98810383+0.j  0.99164854+0.j]

Note how, by virtue of how matrix analytic functions are defined, the Bessel function has acted on the matrix eigenvalues.