eig

Eigenvalues and eigenvectors of symbolic matrix

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Syntax

lambda = eig(A)

[V,D] = eig(A)

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

lambda = eig(B)

[V,D] = eig(B)

Description

example

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

example

[V,D] = eig(A)returns matricesVandD. The columns ofV现在eigenvectors ofA. The diagonal matrixDcontains eigenvalues. If the resultingVhas the same size asA, then the matrixAhas a full set of linearly independent eigenvectors that satisfyA*V = V*D.

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

example

lambda= eig(B)returns numeric eigenvalues of the square variable-precision matrixB. To convert a symbolic matrixAto variable-precision, useB = vpa(A).

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

Examples

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计算特征值

Open Live Script

Compute eigenvalues for the magic square of order 5.

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

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

Compute Numeric Eigenvalues to High Precision

Open Live Script

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

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

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

Convert Symbolic Eigenvalues to Floating-Point

Open Live Script

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

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

lambda =
                       
                         $\begin{array}{l} (\begin{array}{c} root (σ_{1}, z, 1) \\ root (σ_{1}, z, 2) \\ root (σ_{1}, z, 3) \\ root (σ_{1}, z, 4) \\ root (σ_{1}, z, 5) \end{array}) \\ where \\ σ_{1} = z^{5} - 100 z^{4} + 134 z^{3} + 66537 z^{2} - 450198 z - 1294704 \end{array}$

Theeig函数无法找到确切的术语的特征值s of symbolic numbers. Instead, it returns them in terms of therootfunction.

Usevpato numerically approximate the eigenvalues.

lambdaVpa = vpa(lambda)

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

计算特征值and Eigenvectors

Open Live Script

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

A = sym(gallery(5))

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

[v,lambda] = eig(A)

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

lambda =
                       
                         $(\begin{array}{c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array})$

Input Arguments

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`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 matrixAto variable-precision, useB = vpa(A).

Output Arguments

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`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 ofA.

`D`— Eigenvalues (returned as matrix)
symbolic diagonal matrix

Eigenvalues, returned as a symbolic diagonal matrix with the eigenvalues ofAon 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

expand all

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

Behavior changed in R2021b

Ifeig(A)cannot find the exact eigenvalues in terms of symbolic numbers, it now returns the exact eigenvalues in terms of therootfunction 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. Theeigfunction returns the exact eigenvalues in terms of therootfunction. This is consistent with the results returned by thesolveorrootfunction 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)

Usevpato numerically approximate the eigenvalues.

lambdaVpa = vpa(lambda)

lambdaVpa = -2.181032364984695108354692701065 9.8395828502812312578803604206392 -25.131641669799891607267584639192 26.341617610275869035465716505806 91.131473574227486422276200413812

eig

Syntax

Description

Examples

计算特征值

Compute Numeric Eigenvalues to High Precision

Convert Symbolic Eigenvalues to Floating-Point

计算特征值and Eigenvectors

Input Arguments

`A`—Square matrix
symbolic matrix

`B`—Square variable-precision matrix
numeric matrix

Output Arguments

`lambda`— Eigenvalues (returned as vector)
symbolic column vector | column vector of symbolic numbers

`V`— Right eigenvectors
square symbolic matrix

`D`— Eigenvalues (returned as matrix)
symbolic diagonal matrix

`P`— Vector of indices
symbolic row vector

Limitations

Version History

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

See Also

Topics

eig

Syntax

Description

Examples

计算特征值

Compute Numeric Eigenvalues to High Precision

Convert Symbolic Eigenvalues to Floating-Point

计算特征值and Eigenvectors

Input Arguments

A—Square matrixsymbolic matrix

B—Square variable-precision matrixnumeric matrix

Output Arguments

lambda— Eigenvalues (returned as vector)symbolic column vector | column vector of symbolic numbers

V— Right eigenvectorssquare symbolic matrix

D— Eigenvalues (returned as matrix)symbolic diagonal matrix

P— Vector of indicessymbolic row vector

Limitations

Version History

R2021b:eig(A)returns eigenvalues in terms of therootfunction

See Also

Topics

`A`—Square matrix
symbolic matrix

`B`—Square variable-precision matrix
numeric matrix

`lambda`— Eigenvalues (returned as vector)
symbolic column vector | column vector of symbolic numbers

`V`— Right eigenvectors
square symbolic matrix

`D`— Eigenvalues (returned as matrix)
symbolic diagonal matrix

`P`— Vector of indices
symbolic row vector

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