Linear Algebra, 2020a

248 Chapter Three. Maps Between Spaces
3.11 Example Here the 3×3 identity leaves its multiplicand unchanged both from
the left ⎛
⎜
1 0 0
⎞ ⎛ ⎞ ⎛ ⎞
2 3 6 2 3 6
⎟ ⎜ ⎟ ⎜ ⎟
⎝0 1 0⎠
⎝ 1 3 8⎠ = ⎝ 1 3 8⎠
0 0 1 −7 1 0 −7 1 0
and from the right.
⎛ ⎞ ⎛
2 3 6
⎜ ⎟ ⎜
1 0 0
⎞ ⎛ ⎞
2 3 6
⎟ ⎜ ⎟
⎝ 1 3 8⎠
⎝0 1 0⎠ = ⎝ 1 3 8⎠
−7 1 0 0 0 1 −7 1 0
In short, an identity matrix is the identity element of the set of n×n matrices
with respect to the operation of matrix multiplication.
We can generalize the identity matrix by relaxing the ones to arbitrary reals.
The resulting matrix rescales whole rows or columns.
3.12 Definition A diagonal matrix is square and has 0’s off the main diagonal.
⎛
⎞
a 1,1 0 ... 0
0 a 2,2 ... 0
⎜
⎝
⎟
.
⎠
0 0 ... a n,n
3.13 Example From the left, the action of multiplication by a diagonal matrix is
to rescales the rows.
( )(
) (
)
2 0 2 1 4 −1 4 2 8 −2
=
0 −1 −1 3 4 4 1 −3 −4 −4
From the right such a matrix rescales the columns.
( ) ⎛ ⎞
3 0 0 ( )
1 2 1 ⎜ ⎟ 3 4 −2
⎝ 0 2 0 ⎠ =
2 2 2
6 4 −4
0 0 −2
We can also generalize identity matrices by putting a single one in each row
and column in ways other than putting them down the diagonal.
3.14 Definition A permutation matrix is square and is all 0’s except for a single 1
in each row and column.