Python Scipy Linalg Eigh

In this Python Scipy tutorial, we will learn, “Python Scipy Linag Eigh” which is used to compute the eigenvalues or eigenvectors of the hermitian or real symmetric matrix.

• Python Scipy Linalg Eigh
• Python Scipy Linalg Eigh Tridiagonal
• Python Scipy Linalg Eigh Egvals Only
• Python Scipy Linalg Eigh Return

Python Scipy Linalg Eigh

The method eigh() of Python Scipy that exists in a module scipy.linalg for a complex Hermitian or real symmetric matrix, to resolve a conventional or generalized eigenvalue problem.

The syntax is given below.

scipy.linalg.eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, type=1, check_finite=True, subset_by_index=None, subset_by_value=None, driver=None)

Where parameters are:

• a(array_data): The eigenvalues and eigenvectors of a complex Hermitian or real symmetric matrix will be calculated.
• b(array_data): a real symmetric, complex Hermitian, positive matrix. The identity matrix is presumed if missing.
• lower(boolean): if the relevant array data is drawn from the lower or the upper triangle from a and, if applicable, b.
• eigvals_only(boolean): if just eigenvalues should be calculated and not eigenvectors (by default both are computed).
• overwrite_a(boolean): Whether or not to replace data in a.
• overwrite_b(boolean): Whether or not to replace data in b.
• subset_by_index(iterable): The start and end indices of the desired eigenvalues are defined by this two-element iterable if one is provided (ascending order and 0-indexed). [1, 4] is used to only return the second through fifth smallest eigenvalues. The largest three are returned by [n-3, n-1]. exclusive to “evr,” “evx,” and “gvx” drivers. By using int, the entries are immediately transformed to integers ().
• subse_by_value(iterable): This two-element iterable, if given, specifies the half-open interval (a, b) within which, if any, only the eigenvalues between these values are returned. exclusive to “evr,” “evx,” and “gvx” drivers. For the unconstrained endpoints, use np.inf.
• driver(string): Specifies the appropriate LAPACK driver to use. For ordinary issues, you can choose from “ev”, “evd,” “evr,” or “evx,” whereas for generalised problems (where b is not None), you can choose from “gv,” “gvd,” or “gvx.”. “evr” is the default value for common problems. “gvd” is used for entire sets in generalised problems while “gvx” is used for subset requested circumstances.
• check_finite(boolean): if it is necessary to verify that the input matrices only contain finite numbers. Disabling may improve performance, but if the inputs do contain infinities or NaNs, it may cause issues (crashes, non-termination).
• type(int): This parameter identifies the specific kind of problem that needs to be solved for w and v for the generalised problems.

The method eigh() returns two values w (Each of the N (1=N=M) selected eigenvalues was repeated a certain number of times) and v (If eigvals_only is False) of type ndarray.

Let’s take compute the eigenvalue or eigenvectors of complex Hermitian or real symmetric matrix by following the below steps:

Import the required methods or libraries using the below python code.

from scipy import linalg
import numpy as np

Create a matrix, so here we are taking the USA district’s first four-digit pin code out of 5 for a specific area as data values for the matrix, like the pin code of USA district Aleutian East = 99553, Autauga = 36003, Clinton = 45390 and Brooks = 31720.

usa_dist_pin = np.array([[9, 9, 5, 5], [3, 6, 0, 0], [4, 5, 3, 9], [3, 1, 7, 2]])

Now pass the above matrix to the method eigh() for computing the eigenvalues and eigenvectors using the below code.

w_, v_ = linalg.eigh(usa_dist_pin)
print("Eigenvalues",w_)
print("Eigenvectors",v_)

From the above output, this is how to compute the eigenvalues and eigenvector using the method eigh() of Python Scipy.

Read Python Scipy Distance Matrix

Python Scipy Linalg Eigh Tridiagonal

The method eigh_tridiagonal() of Python Scipy in a module scipy.linalg is used for a real symmetric tridiagonal matrix, to find the eigenvalues.

The syntax is given below.

scipy.linalg.eigh_tridiagonal(d, e, select='a', select_range=None, check_finite=True, tol=0.0, lapack_driver='auto')

Where parameters are:

• d(ndarray): The array’s diagonal elements.
• e(ndarray): The array’s off-diagonal elements.
• select_range(min, max): Selection of an eigenvalue range.
• select(v, i , a): Which eigenvalues should be computed, a for all eigenvalues, i for eigenvalues with min <= i <= max indices and v for eigenvalues within the range [(min, max].
• check_finite(boolean): Whether or not to make sure the input matrix only has finite numbers. Disabling may improve performance, but if the inputs do contain infinities or NaNs, it may cause issues (crashes, non-termination).
• tol(float): The strictest tolerance that must be met for each eigenvalue (used exclusively when’stebz’ is the lapack driver). If an eigenvalue (or cluster) falls inside this range, it is said to have converged. If = 0. (the default), the value eps*|a| is used, where |a| is the matrix a’s 1-norm and eps is the machine precision.
• lapack_driver(string): To utilise an LAPACK function, choose from “auto,” “stemr,” “stebz,” “sterf,” or “stev.” When select=’a’, it will use’stemr’; if select=’stebz,’ it will use ‘auto’ (the default). A second LAPACK call (to?STEIN) is made to get the matching eigenvectors when “stebz” is used to determine the eigenvalues and eigvals only=False. Only when eigvals only=True and select=’a’ can the keyword “sterf” be used. Only when select=’a’ may ‘stev’ be used.

The method eigh_tridiagonal() returns w (The eigenvalues are repeated in increasing sequence according to their multiplicity) and v(The column v[:,i] contains the normalized eigenvector that corresponds to the eigenvalue w[i]).

Let’s take an example by following the below steps:

Import the required methods or libraries using the below python code.

from scipy import linalg
import numpy as np

Create diagonal and off-diagonal elements of an array using the below code.

diag_ele = 3*np.ones(6)
off_diag_ele = -1*np.ones(5)

Pass the above arrays to the method eigh_tridiagonal() using the below code.

w_, v_ = linalg.eigh_tridiagonal(diag_ele, off_diag_ele)
print("Eigenvalues",w_)
print("Eigenvectors",v_)

From the output, we can see that this is how to find the eigenvalues for a real symmetric tridiagonal matrix using the method eigh_tridiagonal() of Python Scipy.

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Python Scipy Linalg Eigh Eigvals Only

The parameter eigvals_only is accepted by the method eigh() of Python Scipy to compute the eigenvalues, if this parameter is set to True.

Let’s take the same example that we have used in the above subsection “Python Scipy Linalg Eigh”.

Import the required methods or libraries using the below python code.

from scipy import linalg
import numpy as np

Create a matrix, so here we are taking the USA district’s first four-digit pin code out of 5 for a specific area as data values for the matrix, like the pin code of the USA district Aleutian East = 99553, Autauga = 36003, Clinton = 45390 and Brooks = 31720.

usa_dist_pin = np.array([[9, 9, 5, 5], [3, 6, 0, 0], [4, 5, 3, 9], [3, 1, 7, 2]])

Now pass the above matrix to the method eigh() with parameter eigvals_only equal to True for computing the eigenvalues only using the below code.

linalg.eigh(usa_dist_pin,eigvals_only=True)

From the output, we can see the eigenvalues of the matrix usa_dist_pin using the method eigh() with parameter eigvals_only equal to True of Python Scipy

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Python Scipy Linalg Eigh Return

The method eigh() of Python Scipy that we have learned in the above subsection, returns 3 values a w (Each of the N (1=N=M) selected eigenvalues was repeated a certain number of times) , v (If eigvals_only is False) of type ndarray and raises an error LinAlgError (if an error happened during the computation of the eigenvalues, the b matrix is not definite positive or both. Keep in mind that incorrect results will occur even if there is no error notified if the input matrices are not symmetric or Hermitian).

Here we will see the error part because we have already learned about the w and v values in the above subsection Python Scipy Linalg Eigh.

Import the required methods or libraries using the below python code.

from scipy import linalg
import numpy as np

Create a matrix a and b using the below code.

a = np.array([[9, 9, 5, 5], [3, 6, 0, 0], [4, 5, 3, 9], [3, 1, 7, 2]])
b =np.array([[-9, -9, -5, -5], [-3, -6, 0, 0], [4, -5, 3, -9], [3, -1, -7, 2]])

Pass the above two matrices to the method eigh() using the below code.

w_, v_ = linalg.eigh(a,b)
print("Eigenvalues",w_)
print("Eigenvectors",v_)

The output shows the error LinAlgError which says the matrix b is not a positive matrix.

So in this tutorial, we have learned how to compute the eigenvalues or eigenvectors using the eigh() of Python Scipy. And also know about the error returned by this method with the following topics.

• Python Scipy Linalg Eigh
• Python Scipy Linalg Eigh Tridiagonal
• Python Scipy Linalg Eigh Egvals Only
• Python Scipy Linalg Eigh Return

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• Python Scipy Linalg Eig
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