# eigshow, week 3

An option in `eigshow` demonstrates SVD, the matrix singular value decomposition. The emphasis is on orthogonality.

### Contents

#### The default matrix

This is the third posting about one of my favorite MATLAB programs, `eigshow`. The first two postings were about the `'eig'` option for eigenvalues. This posting is about the `'svd'` option for the singular value decomposition.

The default matrix, shown in the figure window title, is

format rat, format compact A = [1 3; 4 2]/4

A = 1/4 3/4 1 1/2

With the `'svd'` option, you have two green vectors, `x` and `y`, that are perpendicular to each other and that have unit length. They remain perpendicular as you move `x` around the unit circle with the mouse, so they always form an orthonormal basis for two-dimensional space.

The two blue vectors, `Ax` and `Ay`, are images of `x` and `y` under the mapping induced by the matrix `A`. The angle between them changes and their lengths change as you move around the space. The instructions at the bottom of the frame say

`Make A*x perpendicular to A*y`

#### The SVD

When you reach the configuration where not only `x` is perpendicular to `y` but also `Ax` is perpendicular to `Ay`, you have found the singular value decomposition. The SVD is usually described as the matrix factorization

$$ A = U \Sigma V^T $$

with orthogonal $U$ and $V$ and diagonal $\Sigma$. This can be written

$$ A V = U \Sigma $$

For each column of $V$ and the corresponding column of $U$,

$$ A v_j = \sigma_j u_j $$

We can see that when `Ax` is perpendicular to `Ay`, the vectors `x` and `y` are the columns of $V$ and the vectors `Ax` and `Ay` are the columns of $U \Sigma$. The blue curve, which is the image of the unit circle under the transformation induced by $A$, is an ellipse. The vectors `Ax` and `Ay` are its major and minor axes and the half lengths of these axes are the singular values of the matrix.

For this first matrix it looks like the length of `Ax` might be about 1.25 and the length of `Ay` might be about 0.50. We can check this by actually computing the singular values.

```
format short
sigma = svd(A)
```

sigma = 1.2792 0.4886

#### Existence

In our previous posts about eigenvalues, we found that real eigenvectors don't always exist. If the eigenvectors are complex, `eigshow` can never find them. And if the matrix is defective, `eigshow` finds only one eigenvector, not the usual two. The situation is quite different for SVD. The real singular value decomposition always exists. It is always possible to rotate `x` and `y` to a position that makes `Ax` orthogonal to `Ay`.

In the language of abstract linear algebra, we say that the green vectors are an orthonormal basis for the *domain* of the transformation and, if the blue vectors were scaled to have unit length, they would be an orthonormal basis for the *range* of the transformation.

#### A defective matrix

Our final matrix in the eigenvalues post was defective -- it had only one eigenvector. But its SVD is obtained with no difficulty.

```
format rat
A = [6 4; -1 2]/4
```

A = 3/2 1 -1/4 1/2

At first it looks like the final blue vectors happen to be parallel to the display axes. This would say that the matrix $U$ in the SVD is the identity matrix. But a careful look at the final result shows slight stair-stepping in `Ax` and `Ay`. This is confirmed by actually computing the full SVD and observing that $U$ is not quite the identity.

```
format short
[U,S,V] = svd(A)
```

U = -0.9991 -0.0424 -0.0424 0.9991 S = 1.8042 0 0 0.5542 V = -0.8247 -0.5655 -0.5655 0.8247

#### A singular matrix

Our singular example turns out to be a little tricky because the blue ellipse collapses to a one-dimensional line segment. It is hard to tell when `Ax` is perpendicular to `Ay`. It turns out to be when one of them is the zero vector; in other words, when `x` or `y` is the null vector of `A`. When one of `Ax` or `Ay` reaches zero, the other reaches its maximum length, the half-length of the line segment. This length is the nonzero singular value.

```
format rat
A = [2 4; 2 4]/4
```

A = 1/2 1 1/2 1

**Category:**- Eigenvalues,
- Matrices

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