1.5 Diagonal matrices and identity matrices

A diagonal matrix is one that is square, meaning number of rows equals number of columns, and it has 0s on the off-diagonal and non-zeros on the diagonal. In R, you form a diagonal matrix with the diag() function:

diag(1,3) #put 1 on diagonal of 3x3 matrix

     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1

diag(2, 3) #put 2 on diagonal of 3x3 matrix

     [,1] [,2] [,3]
[1,]    2    0    0
[2,]    0    2    0
[3,]    0    0    2

diag(1:4) #put 1 to 4 on diagonal of 4x4 matrix

     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    2    0    0
[3,]    0    0    3    0
[4,]    0    0    0    4

The diag() function can also be used to replace elements on the diagonal of a matrix:

A = matrix(3, 3, 3)
diag(A) = 1
A

     [,1] [,2] [,3]
[1,]    1    3    3
[2,]    3    1    3
[3,]    3    3    1

A = matrix(3, 3, 3)
diag(A) = 1:3
A

     [,1] [,2] [,3]
[1,]    1    3    3
[2,]    3    2    3
[3,]    3    3    3

A = matrix(3, 3, 4)
diag(A[1:3, 2:4]) = 1
A

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

The diag() function is also used to get the diagonal of a matrix.

A = matrix(1:9, 3, 3)
diag(A)

[1] 1 5 9

The identity matrix is a special kind of diagonal matrix with 1s on the diagonal. It is denoted \(\mathbf{I}\). \(\mathbf{I}_3\) would mean a \(3 \times 3\) diagonal matrix. A identity matrix has the property that \(\mathbf{A}\mathbf{I}=\mathbf{A}\) and \(\mathbf{I}\mathbf{A}=\mathbf{A}\) so it is like a 1.

A = matrix(1:9, 3, 3)
I = diag(3)  #shortcut for 3x3 identity matrix
A %*% I

     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9