matrix multiplication

There are two signs: \(*\) and %*% , both means multiplication in broad terms, but they are different when dealing with matrices.

Suppose I have two vectors:

A <- matrix (1:6, nrow = 2)
B <- matrix (5:10, ncol = 2)

View these matrices:

##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6

##      [,1] [,2]
## [1,]    5    8
## [2,]    6    9
## [3,]    7   10

Generally, when we want matrix multiplication

for example, the resulting matrix should have [1,1] as:

\[ 1 \times 5 + 3 \times 6 + 5 \times 7 = 58\] or the [2,2] position be: \[ 2\times 8 + 4 \times 9 + 6 \times 10 = 112\]

We can do this by using the symbol: %*%

A %*% B

##      [,1] [,2]
## [1,]   58   85
## [2,]   76  112

Then what about the symbol: \(*\)

We can create another two column matrices, C and D:

C <- matrix (1:4, ncol = 1)
D <- matrix (5:8, ncol = 1)

View C and D:

##      [,1]
## [1,]    1
## [2,]    2
## [3,]    3
## [4,]    4

##      [,1]
## [1,]    5
## [2,]    6
## [3,]    7
## [4,]    8

Let’s try to use \(*\)

C * D

##      [,1]
## [1,]    5
## [2,]   12
## [3,]   21
## [4,]   32

As can be seen, this \(*\) symbol can be regarded as: computing the dot product

while for %*% it means: give me the cross product