eig

Eigenvalues and eigenvectors of symbolic matrix

Syntax

lambda = eig(A)

[V,D] = eig(A)

[V,D,P] = eig(A)

lambda = eig(B)

[V,D] = eig(B)

Description

lambda = eig(A) returns a symbolic vector containing the eigenvalues of the square symbolic matrix A.

[V,D] = eig(A) returns matrices V and D. The columns of V present eigenvectors of A. The diagonal matrix D contains eigenvalues. If the resulting V has the same size as A, then the matrix A has a full set of linearly independent eigenvectors that satisfy A*V = V*D.

[V,D,P] = eig(A) returns a vector of indices P. The length of P is equal to the number of linearly independent eigenvectors, so that A*V = V*D(P,P).

lambda = eig(B) returns numeric eigenvalues of the square variable-precision matrix B. To convert a symbolic matrix A to variable-precision, use B = vpa(A).

[V,D] = eig(B) also returns numeric eigenvectors.

Examples

Compute Eigenvalues

Compute eigenvalues for the magic square of order 5.

A = sym(magic(5));
lambda = eig(A)

lambda = 
$$\left(\begin{array}{c}65\\ \sqrt{\frac{625}{2}-\frac{5\hspace{0.17em}\sqrt{3145}}{2}}\\ \sqrt{\frac{5\hspace{0.17em}\sqrt{3145}}{2}+\frac{625}{2}}\\ -\sqrt{\frac{625}{2}-\frac{5\hspace{0.17em}\sqrt{3145}}{2}}\\ -\sqrt{\frac{5\hspace{0.17em}\sqrt{3145}}{2}+\frac{625}{2}}\end{array}\right)$$

Compute Numeric Eigenvalues to High Precision

Compute numeric eigenvalues for the magic square of order 5 using variable-precision arithmetic.

A = magic(5);
lambda = eig(vpa(A))

lambda = 
$$\left(\begin{array}{c}65.0\\ 21.276765471473795530626426697974\\ 13.126280930709218802525643085949\\ -13.126280930709218802525643085949\\ -21.276765471473795530626426697974\end{array}\right)$$

Convert Symbolic Eigenvalues to Floating-Point

Create a 5-by-5 symbolic matrix from the magic square of order 6. Compute the eigenvalues of the matrix using eig.

M = magic(6);
A = sym(M(1:5,1:5));
lambda = eig(A)

lambda = 
$$\begin{array}{l}\left(\begin{array}{c}\mathrm{root}\left({\sigma }_1,z,1\right)\\ \mathrm{root}\left({\sigma }_1,z,2\right)\\ \mathrm{root}\left({\sigma }_1,z,3\right)\\ \mathrm{root}\left({\sigma }_1,z,4\right)\\ \mathrm{root}\left({\sigma }_1,z,5\right)\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=z^5-100\hspace{0.17em}{z}^4+134\hspace{0.17em}{z}^3+66537\hspace{0.17em}{z}^2-450198\hspace{0.17em}z-1294704\end{array}$$

The eig function cannot find the exact eigenvalues in terms of symbolic numbers. Instead, it returns them in terms of the root function.

Use vpa to numerically approximate the eigenvalues.

lambdaVpa = vpa(lambda)

lambdaVpa = 
$$\left(\begin{array}{c}-2.181032364984695108354692701065\\ 9.8395828502812312578803604206392\\ -25.131641669799891607267584639192\\ 26.341617610275869035465716505806\\ 91.131473574227486422276200413812\end{array}\right)$$

Compute Eigenvalues and Eigenvectors

Compute the eigenvalues and eigenvectors for one of the MATLAB® test matrices.

A = sym(gallery(5))

A = 
$$\left(\begin{array}{ccccc}-9& 11& -21& 63& -252\\ 70& -69& 141& -421& 1684\\ -575& 575& -1149& 3451& -13801\\ 3891& -3891& 7782& -23345& 93365\\ 1024& -1024& 2048& -6144& 24572\end{array}\right)$$

[v,lambda] = eig(A)

v = 
$$\left(\begin{array}{c}0\\ \frac{21}{256}\\ -\frac{71}{128}\\ \frac{973}{256}\\ 1\end{array}\right)$$

lambda = 
$$\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$$

Input Arguments

A — Square matrix

symbolic matrix

Square matrix, specified as a symbolic matrix.

B — Square variable-precision matrix

numeric matrix

Square variable-precision matrix, specified as a numeric matrix. To convert a symbolic matrix A to variable-precision, use B = vpa(A).

Output Arguments

lambda — Eigenvalues (returned as vector)

symbolic column vector | column vector of symbolic numbers

Eigenvalues, returned as a symbolic column vector or column vector of symbolic numbers.

V — Right eigenvectors

square symbolic matrix

Right eigenvectors, returned as a square symbolic matrix whose columns are the right eigenvectors of A.

D — Eigenvalues (returned as matrix)

symbolic diagonal matrix

Eigenvalues, returned as a symbolic diagonal matrix with the eigenvalues of A on the main diagonal.

P — Vector of indices

symbolic row vector

Vector of indices, returned as a symbolic row vector whose length is the total number of linearly independent eigenvectors.

Limitations

Matrix computations involving many symbolic variables can be slow. To increase the computational speed, reduce the number of symbolic variables by substituting the given values for some variables.

Version History

Introduced before R2006a

R2021b: eig(A) returns eigenvalues in terms of the root function

Behavior changed in R2021b

If eig(A) cannot find the exact eigenvalues in terms of symbolic numbers, it now returns the exact eigenvalues in terms of the root function instead. In previous releases, eig(A) returns the eigenvalues as floating-point numbers.

For example, compute the eigenvalues of a 5-by-5 symbolic matrix. The eig function returns the exact eigenvalues in terms of the root function. This is consistent with the results returned by the solve or root function when solving for the roots of a polynomial.

M = magic(6);
A = sym(M(1:5,1:5));
lambda = eig(A)

lambda = 
root(z^5 - 100*z^4 + 134*z^3 + 66537*z^2 - 450198*z - 1294704, z, 1)
root(z^5 - 100*z^4 + 134*z^3 + 66537*z^2 - 450198*z - 1294704, z, 2)
root(z^5 - 100*z^4 + 134*z^3 + 66537*z^2 - 450198*z - 1294704, z, 3)
root(z^5 - 100*z^4 + 134*z^3 + 66537*z^2 - 450198*z - 1294704, z, 4)
root(z^5 - 100*z^4 + 134*z^3 + 66537*z^2 - 450198*z - 1294704, z, 5)

Use vpa to numerically approximate the eigenvalues.

lambdaVpa = vpa(lambda)

lambdaVpa =
 -2.181032364984695108354692701065
 9.8395828502812312578803604206392
-25.131641669799891607267584639192
 26.341617610275869035465716505806
 91.131473574227486422276200413812