# Linear Algebra (`scipy.linalg`)¶

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.

## scipy.linalg vs numpy.linalg¶

`scipy.linalg` contains all the functions in `numpy.linalg`.
plus some other more advanced ones not contained in `numpy.linalg`

Another advantage of using `scipy.linalg` over `numpy.linalg` is that
it is always compiled with BLAS/LAPACK support, while for numpy this is
optional. Therefore, the scipy version might be faster depending on how
numpy was installed.

Therefore, unless you don’t want to add `scipy` as a dependency to
your `numpy` program, use `scipy.linalg` instead of `numpy.linalg`

## numpy.matrix vs 2D numpy.ndarray¶

The classes that represent matrices, and basic operations such as
matrix multiplications and transpose are a part of `numpy`.
For convenience, we summarize the differences between `numpy.matrix`
and `numpy.ndarray` here.

`numpy.matrix` is matrix class that has a more convenient interface
than `numpy.ndarray` for matrix operations. This class supports for
example MATLAB-like creation syntax via the, has matrix multiplication
as default for the `*` operator, and contains `I` and `T` members
that serve as shortcuts for inverse and transpose:

```
>>> import numpy as np
>>> A = np.mat('[1 2;3 4]')
>>> A
matrix([[1, 2],
[3, 4]])
>>> A.I
matrix([[-2. , 1. ],
[ 1.5, -0.5]])
>>> b = np.mat('[5 6]')
>>> b
matrix([[5, 6]])
>>> b.T
matrix([[5],
[6]])
>>> A*b.T
matrix([[17],
[39]])
```

Despite its convenience, the use of the `numpy.matrix` class is
discouraged, since it adds nothing that cannot be accomplished
with 2D `numpy.ndarray` objects, and may lead to a confusion of which class
is being used. For example, the above code can be rewritten as:

```
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
[3, 4]])
>>> linalg.inv(A)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> b = np.array([[5,6]]) #2D array
>>> b
array([[5, 6]])
>>> b.T
array([[5],
[6]])
>>> A*b #not matrix multiplication!
array([[ 5, 12],
[15, 24]])
>>> A.dot(b.T) #matrix multiplication
array([[17],
[39]])
>>> b = np.array([5,6]) #1D array
>>> b
array([5, 6])
>>> b.T #not matrix transpose!
array([5, 6])
>>> A.dot(b) #does not matter for multiplication
array([17, 39])
```

`scipy.linalg` operations can be applied equally to
`numpy.matrix` or to 2D `numpy.ndarray` objects.

## 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

```
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2],[3,4]])
array([[1, 2],
[3, 4]])
>>> linalg.inv(A)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> A.dot(linalg.inv(A)) #double check
array([[ 1.00000000e+00, 0.00000000e+00],
[ 4.44089210e-16, 1.00000000e+00]])
```

### 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:

```
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
[3, 4]])
>>> b = np.array([[5],[6]])
>>> b
array([[5],
[6]])
>>> linalg.inv(A).dot(b) #slow
array([[-4. ],
[ 4.5]]
>>> A.dot(linalg.inv(A).dot(b))-b #check
array([[ 8.88178420e-16],
[ 2.66453526e-15]])
>>> np.linalg.solve(A,b) #fast
array([[-4. ],
[ 4.5]])
>>> A.dot(np.linalg.solve(A,b))-b #check
array([[ 0.],
[ 0.]])
```

### 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:

```
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
[3, 4]])
>>> linalg.det(A)
-2.0
```

### 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 .

Examples:

```
>>> import numpy as np
>>> from scipy import linalg
>>> A=np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
[3, 4]])
>>> linalg.norm(A)
5.4772255750516612
>>> linalg.norm(A,'fro') # frobenius norm is the default
5.4772255750516612
>>> linalg.norm(A,1) # L1 norm (max column sum)
6
>>> linalg.norm(A,-1)
4
>>> linalg.norm(A,inf) # L inf norm (max row sum)
7
```

### 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()
```

### 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 addition, `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.

```
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2],[3,4]])
>>> la,v = linalg.eig(A)
>>> l1,l2 = la
>>> print l1, l2 #eigenvalues
(-0.372281323269+0j) (5.37228132327+0j)
>>> print v[:,0] #first eigenvector
[-0.82456484 0.56576746]
>>> print v[:,1] #second eigenvector
[-0.41597356 -0.90937671]
>>> print np.sum(abs(v**2),axis=0) #eigenvectors are unitary
[ 1. 1. ]
>>> v1 = np.array(v[:,0]).T
>>> print linalg.norm(A.dot(v1)-l1*v1) #check the computation
3.23682852457e-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 addition, 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` .

```
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2,3],[4,5,6]])
>>> A
array([[1, 2, 3],
[4, 5, 6]])
>>> M,N = A.shape
>>> U,s,Vh = linalg.svd(A)
>>> Sig = linalg.diagsvd(s,M,N)
>>> U, Vh = U, Vh
>>> U
array([[-0.3863177 , -0.92236578],
[-0.92236578, 0.3863177 ]])
>>> Sig
array([[ 9.508032 , 0. , 0. ],
[ 0. , 0.77286964, 0. ]])
>>> Vh
array([[-0.42866713, -0.56630692, -0.7039467 ],
[ 0.80596391, 0.11238241, -0.58119908],
[ 0.40824829, -0.81649658, 0.40824829]])
>>> U.dot(Sig.dot(Vh)) #check computation
array([[ 1., 2., 3.],
[ 4., 5., 6.]])
```

[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 `linalg.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]]
```

### Interpolative Decomposition¶

`scipy.linalg.interpolative` contains routines for computing the
interpolative decomposition (ID) of a matrix. For a matrix of rank
this is a factorization

where is a permutation matrix with
, i.e., . This can equivalently be written as ,
where and
are the *skeleton* and *interpolation matrices*, respectively.

See also

`scipy.linalg.interpolative` — for more information.

## 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.

## Special matrices¶

SciPy and NumPy provide several functions for creating special matrices that are frequently used in engineering and science.

Type | Function | Description |
---|---|---|

block diagonal | scipy.linalg.block_diag |
Create a block diagonal matrix from the provided arrays. |

circulant | scipy.linalg.circulant |
Construct a circulant matrix. |

companion | scipy.linalg.companion |
Create a companion matrix. |

Hadamard | scipy.linalg.hadamard |
Construct a Hadamard matrix. |

Hankel | scipy.linalg.hankel |
Construct a Hankel matrix. |

Hilbert | scipy.linalg.hilbert |
Construct a Hilbert matrix. |

Inverse Hilbert | scipy.linalg.invhilbert |
Construct the inverse of a Hilbert matrix. |

Leslie | scipy.linalg.leslie |
Create a Leslie matrix. |

Pascal | scipy.linalg.pascal |
Create a Pascal matrix. |

Toeplitz | scipy.linalg.toeplitz |
Construct a Toeplitz matrix. |

Van der Monde | numpy.vander |
Generate a Van der Monde matrix. |

For examples of the use of these functions, see their respective docstrings.