# eig

Eigenvalues and eigenvectors of symbolic matrix

## Syntax

``lambda = eig(A)``
``[V,D] = eig(A)``
``[V,D,p] = eig(A)``

## Description

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

example

````[V,D] = eig(A)` returns the eigenvectors and eigenvalues of `A` as symbolic matrices `V` and `D`. The columns of `V` present eigenvectors of `A`. The main diagonal of `D` present eigenvalues of `A`. If `V` is the same size as `A`, then the matrix `A` has a full set of linearly independent eigenvectors that satisfy `A*V = V*D`.If `V` has fewer columns than `A`, then the matrix `A` is defective. In this case, at least one of the eigenvalues λ has an algebraic multiplicity m > 1 with fewer than m linearly independent eigenvectors associated with λ. ```

example

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

## Examples

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Compute eigenvalues for the magic square of order 5.

```A = sym(magic(5)); lambda = eig(A)```
```lambda =  $\left(\begin{array}{c}65\\ \sqrt{\frac{625}{2}-\frac{5 \sqrt{3145}}{2}}\\ \sqrt{\frac{5 \sqrt{3145}}{2}+\frac{625}{2}}\\ -\sqrt{\frac{625}{2}-\frac{5 \sqrt{3145}}{2}}\\ -\sqrt{\frac{5 \sqrt{3145}}{2}+\frac{625}{2}}\end{array}\right)$```

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

```A = magic(5); lambda = eig(vpa(A))```
```lambda =  $\left(\begin{array}{c}65.0\\ 21.276765471473795530626426697974\\ 13.126280930709218802525643085949\\ -13.126280930709218802525643085949\\ -21.276765471473795530626426697974\end{array}\right)$```

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 =  ```

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 the exact eigenvalues and eigenvectors for one of the MATLAB® test matrices that is defective.

`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,D] = eig(A)`
```V =  $\left(\begin{array}{c}0\\ \frac{21}{256}\\ -\frac{71}{128}\\ \frac{973}{256}\\ 1\end{array}\right)$```
```D =  $\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$```

The output `V` is a 5-by-1 column vector with five-fold eigenvalues of 0. This result means that the matrix `A` has an eigenvalue of 0 with algebraic multiplicity 5 and geometric multiplicity 1.

Compute the exact eigenvalues and eigenvectors of a 4-by-4 symbolic matrix. Return a vector of indices that relate the eigenvalues to their linearly independent eigenvectors.

```syms c A = [c 1 0 0; 0 c 0 0; 0 0 3*c 0; 0 0 0 3*c]; [V,D,p] = eig(A)```
```V =  $\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$```
```D =  $\left(\begin{array}{cccc}c& 0& 0& 0\\ 0& c& 0& 0\\ 0& 0& 3 c& 0\\ 0& 0& 0& 3 c\end{array}\right)$```
```p = 1×3 1 3 4 ```

The matrix `A` has two eigenvalues, $\mathit{c}$ and $3\text{\hspace{0.17em}}\mathit{c}$, where each eigenvalue occurs twice. Meanwhile, there are three linearly independent eigenvectors. The vector of indices `p` shows that:

• `p(1) = 1`, so the first eigenvector (the first column of `V`) corresponds to the first diagonal element of `D` with eigenvalue $\mathit{c}$.

• `p(2) = 3`, so the second eigenvector (the second column of `V`) corresponds to the third diagonal element of `D` with eigenvalue $3\text{\hspace{0.17em}}\mathit{c}$.

• `p(3) = 4`, so the third eigenvector (the third column of `V`) corresponds to the fourth diagonal element of `D` with eigenvalue $3\text{\hspace{0.17em}}\mathit{c}$.

This result means the eigenvalue $\mathit{c}$ that occurs twice has only one linearly independent eigenvector (the eigenvalue $\mathit{c}$ has algebraic multiplicity 2 and geometric multiplicity 1). The eigenvalue $3\text{\hspace{0.17em}}\mathit{c}$ that occurs twice has two linearly independent eigenvectors (the eigenvalue $3\text{\hspace{0.17em}}\mathit{c}$ has algebraic multiplicity 2 and geometric multiplicity 2).

Show that the matrix `A*V` is equal to `V*D(p,p)`.

`A*V`
```ans =  $\left(\begin{array}{ccc}c& 0& 0\\ 0& 0& 0\\ 0& 3 c& 0\\ 0& 0& 3 c\end{array}\right)$```
`V*D(p,p)`
```ans =  $\left(\begin{array}{ccc}c& 0& 0\\ 0& 0& 0\\ 0& 3 c& 0\\ 0& 0& 3 c\end{array}\right)$```
`tf = isequal(A*V,V*D(p,p))`
```tf = logical 1 ```

## Input Arguments

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Square matrix, specified as a symbolic matrix.

## Output Arguments

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

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

Eigenvalues, returned as a symbolic diagonal matrix. The eigenvalues of `A` are on the main diagonal of `D`.

Vector of indices, returned as a symbolic row vector. The length of `p` is the total number of linearly independent eigenvectors of `A`.

## Tips

• 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.

• Calling `eig` for numeric matrices that are not symbolic objects (not created by `sym`, `syms`, or `vpa`) invokes the MATLAB® `eig` function.

• The symbolic `eig` function does not support solving the generalized eigenvalue problem (with two input arguments). To solve the generalized eigenvalue problem, use the MATLAB `eig` function instead by converting the input matrices to a MATLAB numeric type.

## Version History

Introduced before R2006a

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