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

```{r nomessages, echo = FALSE}
knitr::opts_chunk$set(
  warning = FALSE,
  message = FALSE,
  fig.height = 5,
  fig.width = 5
)
options(digits=4)
par(mar=c(3,3,1,1)+.1)
```

The following examples illustrate the basic properties of the determinant of a matrix.
We do this first with simple numerical examples and then using geometric diagrams.

### Create a 2 x 2 matrix

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

### 1. Interchange two rows or cols changes the sign: -> -1 * det(A)

```{r }
det(A[ 2:1, ])
det(A[, 2:1 ])
```

### 2. transpose -> det (A) unchanged

```{r }
det( t(A) )
```

### 3. multiply row * k -> k * det(A)

Note that to multiply rows by *different* constants requires a diagonal matrix on the left.

```{r }
diag(c(3, 1)) %*% A
det( diag(c(3, 1))  %*% A)
```

### 4. multiply matrix * k -> k^2 * det(A)

This is because multiplying a matrix by a constant multiplies **each** row.

```{r }
det(3 * A)
3^2 * det(A)
```

### 5. det (A  B) ->  det(A) * det(B)

The determinant of a product is the product of the determinants.  The same holds for
any number of terms in a matrix product.

```{r }
B <- matrix(c(4, 2, 
             3, 5), nrow=2, byrow=TRUE)
B
det(A %*% B)
det(A) * det(B)
```

### 6. proportional rows or columns -> `det() == 0`

Here we just add an additional copy of column 1 of a matrix, so `C[,3] == C[,1]`.
The determinant is 0 because the columns are linearly dependent.

```{r }
C <- matrix(c(1, 5, 
             2, 6,
             4, 4), nrow=3, byrow=TRUE)
C <- cbind(C, C[,1])
C
det(C)
```

### 7. Add multiple of one row to another -> det unchanged

This is the principle behind one of the elementary row operations.

```{r }
A[2,] <- A[2,] - 2*A[1,]
det(A)
```

### 8.  Geometric interpretation

Many aspects of matrices and vectors have geometric interpretations.
For $2 \times 2$ matrices, the determinant is the **area** of the parallelogram
defined by the rows (or columns), plotted in a 2D space.
(For $3 \times 3$ matrices, the determinant is the **volume** of a parallelpiped in 3D space.)
```{r}
A <- matrix(c(3, 1, 
              2, 4), nrow=2, byrow=TRUE)
A
det(A)
```
The `matlib` package has some handy functions (`vectors()`) for drawing geometric diagrams.
```{r det-diagram1,fig.width=5, fig.height=5}
#| fig.alt: A diagram showing two vectors, a1 and a2, illustrating that the area of the parallelogram they form is the determinant of the matrix with columns a1 and a2
library(matlib)
xlim <- c(0,6)
ylim <- c(0,6)
par(mar=c(3,3,1,1)+.1)
plot(xlim, ylim, type="n", xlab="X1", ylab="X2", asp=1)
sum <- A[1,] + A[2,]
# draw the parallelogram determined by the rows of A
polygon( rbind(c(0,0), A[1,], sum, A[2,]), col=rgb(1,0,0,.2))
vectors(A, labels=c("a1", "a2"), pos.lab=c(4,2))
vectors(sum, origin=A[1,], col="gray")
vectors(sum, origin=A[2,], col="gray")
# add some annotations
text(0,6, "det(A) is the area of its row vectors", pos=4)
text(mean(A[,1]), mean(A[,2]), "det(A)", cex=1.25)

```

There is a simple [visual proof of this fact about determinants](https://math.stackexchange.com/questions/29128/why-determinant-of-a-2-by-2-matrix-is-the-area-of-a-parallelogram)
but it is easiest to
see in the case of a diagonal matrix, where the row vectors are orthogonal, so area
is just height x width.

```{r}
(D <- 2 * diag(2))
det(D)
```
Plot this as before:
```{r det-diagram2,fig.width=4, fig.height=4}
#| fig.alt: A diagram showing two orthogonal vectors, d1 and d2 (at right angles). The determinant of the matrix containing them is the area of the square they form.
par(mar=c(3,3,1,1)+.1)
plot(c(0,2), c(0,2), type="n", xlab="X1", ylab="X2", asp=1)
sum <- D[1,] + D[2,]
polygon( rbind(c(0,0), D[1,], sum, D[2,]), col=rgb(0,1,0,.2))
vectors(D, labels=c("d1", "d2"), pos.lab=c(3,4))
vectors(sum, origin=D[1,], col="gray")
vectors(sum, origin=D[2,], col="gray")
text(mean(D[,1]), mean(D[,2]), "det(D)", cex=1.25)
```

Finally, we can also see why the determinant is zero when the rows or columns are proportional.

```{r}
(B <- matrix(c(1, 2, 2, 4), 2,2))
det(B)
```
Such vectors are called *collinear*.  They enclose no *area*.
```{r det-diagram3,fig.width=4, fig.height=4}
#| fig-alt: A diagram showing two collinear (proportional) vectors, b1 and b2. The determinant of the matrix containing them is zero.
par(mar=c(3,3,1,1)+.1)
plot(c(0,4), c(0,4), type="n", xlab="X1", ylab="X2", asp=1)
vectors(B, labels=c("b1", "b2"), pos.lab=c(4,2))

```