---
title: "2. Evaluation of determinants"
author: "Michael Friendly"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{2. Evaluation of determinants}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r nomessages, echo = FALSE}
knitr::opts_chunk$set(
  warning = FALSE,
  message = FALSE
)
options(digits=4)
par(mar=c(5,4,1,1)+.1)
```

This example shows two classical ways to find the determinant, $\det(A)$ of a square matrix. They each work
by reducing the problem to a series of smaller ones which can be more easily calculated.

```{r }
	library(matlib)
```

##  1. Calculate `det()` by cofactor expansion

Set up a $3 \times 3$ matrix, and find its determinant (so we know what the answer should be).

```{r }
  A <- matrix(c(4, 2, 1,
                5, 6, 7,
                1, 0, 3), nrow=3, byrow=TRUE)
  det(A)
```

### Find cofactors of row 1 elements

The cofactor $A_{i,j}$ of element $a_{i,j}$ is the signed determinant of what is left when row i, column j
of the matrix $A$ are deleted.  NB: In R, negative subscripts delete rows or columns.

```{r }
  cat(cofactor(A, 1, 1),  "  ==  ",  1 * det( (A[-1, -1]), "\n" ))
  cat(cofactor(A, 1, 2),  "  ==  ", -1 * det( (A[-1, -2]), "\n" ))
  cat(cofactor(A, 1, 3),  "  ==  ",  1 * det( (A[-1, -3]), "\n" ))
```

### det() = product of row with cofactors

In symbols: $\det(A) = a_{1,1} * A_{1,1} + a_{1,2} * A_{1,2} + a_{1,3} * A_{1,3}$

`rowCofactors()` is a convenience function, that calculates these all together

```{r }
  rowCofactors(A, 1)
```

Voila: Multiply row 1 times the cofactors of its elements.  NB: In R, this multiplication gives a $1 \times 1$ matrix.

```{r }
  A[1,] %*% rowCofactors(A, 1)
  all.equal( det(A), c(A[1,] %*% rowCofactors(A, 1)) )
```

##  2. Finding `det()` by Gaussian elimination (pivoting)

This example follows Green and Carroll, Table 2.2.
Start with a 4 x 4 matrix, $M$, and save `det(M)`.

```{r }
  M <- matrix(c(2, 3, 1, 2,
                4, 2, 3, 4,
                1, 4, 2, 2,
                3, 1, 0, 1), nrow=4, ncol=4, byrow=TRUE)
  (dsave <- det(M))

# ### 'pivot' on the leading diagonal element, M[1,1]:
```

`det()` will be the product of the 'pivots', the leading diagonal elements.
This step reduces row 1 and column 1 to 0, so it may be discarded.
NB: In R, dropping a row/column can change a matrix to a vector, so we use
`drop = FALSE` inside the subscript.

```{r }
  (d <- M[1,1])

    #-- Reduce row 1, col 1 to 0
  (M[1,] <- M[1,, drop=FALSE] / M[1, 1])
  (M <- M - M[,1] %*%  M[1,, drop=FALSE])

    #-- Drop first row and column
  M <- M[-1, -1]

    #-- Accumulate the product of pivots
  d <- d * M[1, 1]
```

### Repeat, reducing new row, col 1 to 0

```{r }
  (M[1,] <- M[1,, drop=FALSE] / M[1,1])
  (M <- M - M[,1] %*%  M[1,, drop=FALSE])
  M <- M[-1, -1]
  d = d * M[1, 1]
```

### Repeat once more. d = det(M)

```{r }
  (M[1,] <- M[1,, drop=FALSE] / M[1,1])
  (M <- M - M[,1] %*%  M[1,, drop=FALSE])
  M <- M[-1, -1, drop=FALSE]
  d <- d * M[1, 1]

  # did we get it right?
  all.equal(d, dsave)
```