Using explicit matrices is not practical for some very large problems. Instead, we can use Spot operators. A Spot operator represents a matrix, and can be treated in a similar way, but it doesn't rely on the matrix itself to implement most of the methods. This short guide will show you how to make and use Spot operators.

## Contents

## Creating Operators

Create a new operator using the constructor from the appropriate operator class. For example, to make an operator A that consists of all ones, with three rows and two columns, use the `opOnes` class:

A = opOnes(3,2)

A = Spot operator: Ones(3,2) rows: 3 complex: no cols: 2 type: Ones

We see that Spot displays some information about A when we leave off the semicolon, such as its construction, the number of rows and columns, and whether it is complex. We can also use the `double` method to construct the underlying matrix:

double(A)

ans = 1 1 1 1 1 1

Operators can be easily added or subtracted. For example, make an operator B consisting of all twos and add it to A:

B = 2*opOnes(3,2); C = A + B; double(C)

ans = 3 3 3 3 3 3

If you add a matrix and an operator, Spot will first automatically convert the matrix into a Spot operator using opMatrix. We can discover other information about the operator C by using the methods `size`, `disp`, and `whos`:

```
whos C
```

Name Size Bytes Class Attributes C 3x2 1732 opSum

## Applying Operators

Spot operators can be multiplied by vectors just like MATLAB matrices. Make a vector x and apply C to it:

x = [1;2]; y = C*x

y = 9 9 9

We can also multiply by the adjoint of an operator:

w = [1;2;3]; z = C'*w;

## Subset Assignment and Reference

If we are only interested in applying part of an operator, we can create a new operator that is a restriction of the existing one. The indexing used is the same as in MATLAB matrices; we can specify rows, columns, or individual elements, we can extract rows or columns in reverse, and we can repeat entries:

x = [1;2;3;4;5]; A = opDiag(x); % Create a 5x5 operator with x's values on its diagonal B = A(2:4,:); % Extract rows 2-4 of A double(B)

ans = 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0

We can also assign values to a subset of an operator, again using the same syntax as in MATLAB matrices. The row and column indices we specify don't have to be within those of the original operator. In fact, we can specify indices that don't overlap at all with the original operator, and we will simply end up with a larger operator. Zero out part of A:

A(1:2,:) = 0; double(A)

ans = 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5

Assign a new operator, C, to a subset of B:

C = opOnes(2,2); B(2:3,1:2) = C; double(B)

ans = 0 2 0 0 0 1 1 3 0 0 1 1 0 4 0

## Creating More Complex Operators

Spot operators can be combined into more complex operators using methods such as blkdiag and kron. Whenever a matrix is passed to one of these methods, it is automatically converted to a Spot operator. `blkdiag` takes a list of operators and matrices and creates a block diagonal operator:

A = opOnes(2,2); B = 2*opOnes(3,2); C = 3*opOnes(1,3); D = blkdiag(A, B, C); double(D)

ans = 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 2 2 0 0 0 0 0 2 2 0 0 0 0 0 0 0 3 3 3

We can also have the blocks overlap and create anti-diagonal operators (see "Using the Methods"). Operators can be horizontally or vertically concatenated using opDictionary or opStack, or simply by passing them as elements in a matrix:

E = 4*opOnes(2,3); F = [A E]; double(F)

ans = 1 1 4 4 4 1 1 4 4 4

The `kron` method computes the Kronecker product of an arbitrary number of operators:

G = opMatrix([2 1;3 0]); H = opMatrix([1 2]); K = kron(G, H); double(K)

ans = 2 4 1 2 3 6 0 0

For more information on how to work with Spot operators, see "Using the Methods". For a list of methods, see the "Index of Methods". For a list of the Spot operator classes and what they do, including fast transformations, random ensembles, and convolution, see the "Index of Operators".