knitr::opts_chunk$set( warning = FALSE, message = FALSE, fig.height = 5, fig.width = 5 ) options(digits=4) par(mar=c(5,4,1,1)+.1)

In matrix algebra, the inverse of a matrix is defined only for *square* matrices,
and if a matrix is *singular*, it does not have an inverse.

The **generalized inverse** (or *pseudoinverse*)
is an extension of the idea of a matrix inverse,
which has some but not all the properties of an ordinary inverse.

A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution.

```
library(matlib)
```

Construct a square, *singular* matrix [See: Timm, EX. 1.7.3]

A <-matrix(c(4, 4, -2, 4, 4, -2, -2, -2, 10), nrow=3, ncol=3, byrow=TRUE) det(A)

The rank is 2, so `inv(A)`

wont work

R(A)

In the echelon form, this rank deficiency appears as the final row of zeros

echelon(A)

`inv()`

will throw an error

```
try(inv(A))
```

A **generalized inverse** does exist for any matrix,
but unlike the ordinary inverse, the generalized inverse is not unique, in the
sense that there are various ways of defining a generalized inverse with
various inverse-like properties. The function `matlib::Ginv()`

calculates
a *Moore-Penrose* generalized inverse.

```
(AI <- Ginv(A))
```

We can also view this as fractions:

Ginv(A, fractions=TRUE)

The generalized inverse is defined as the matrix $A^-$ such that

- $A * A^- * A = A$ and
- $A^- * A * A^- = A^-$

A %*% AI %*% A AI %*% A %*% AI

In addition, both $A * A^-$ and $A^- * A$ are symmetric, but
neither product gives an identity matrix, `A %*% AI != AI %*% A != I`

zapsmall(A %*% AI) zapsmall(AI %*% A)

For a *rectangular matrix*, $A^- = (A^{T} A)^{-1} A^{T}$
is the generalized inverse of $A$
if $(A^{T} A)^-$ is the ginv of $(A^{T} A)$ [See: TIMM: EX 1.6.11]

A <- cbind( 1, matrix(c(1, 0, 1, 0, 0, 1, 0, 1), nrow=4, byrow=TRUE)) A

This $4 \times 3$ matrix is not of full rank, because columns 2 and 3 sum to column 1.

R(A) (AA <- t(A) %*% A) (AAI <- Ginv(AA))

The generalized inverse of $A$ is $(A^{T} A)^- A^{T}$, `AAI * t(A)`

AI <- AAI %*% t(A)

Show that it is a generalized inverse:

A %*% AI %*% A AI %*% A %*% AI

**Any scripts or data that you put into this service are public.**

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.