Fundamentals of Matrix Algebra, 2011a

Chapter 3
Operaons on Matrices
.
Ḋefinion 19
Transpose
.
Let A be an m × n matrix. The tranpsose of A, denoted A T ,
is the n × m matrix whose columns are the respecve rows
of A.
Examples will make this definion clear.
. Example 60 Find the transpose of A =
[ ] 1 2 3
.
4 5 6
S Note that A is a 2 × 3 matrix, so A T will be a 3 × 2 matrix. By the
definion, the first column of A T is the first row of A; the second column of A T is the
second row of A. Therefore, ⎡
1
⎤
4
A T = ⎣ 2 5 ⎦ .
.
3 6
. Example 61 Find the transpose of the following matrices.
⎡
A = ⎣ 7 2 9 1 ⎤ ⎡
⎤
1 10 −2
2 −1 3 0 ⎦ B = ⎣ 3 −5 7 ⎦ C = [ 1 −1 7 8 3 ]
−5 3 0 11 4 2 −3
S We find each transpose using the definion without explanaon.
Make note of the dimensions of the original matrix and the dimensions of its transpose.
⎡ ⎤
⎡
⎤
7 2 −5 ⎡
⎤ 1
A T = ⎢ 2 −1 3
1 3 4
−1
⎥
⎣ 9 3 0 ⎦ BT = ⎣ 10 −5 2 ⎦ C T =
⎢ 7
⎥
−2 7 −3 ⎣ 8 ⎦
1 0 11
.
3
Noce that with matrix B, when we took the transpose, the diagonal did not change.
We can see what the diagonal is below where we rewrite B and B T with the diagonal in
bold. We’ll follow this by a definion of what we mean by “the diagonal of a matrix,”
along with a few other related definions.
⎡
⎤ ⎡
1 10 −2
B = ⎣ 3 –5 7 ⎦ B T = ⎣ 1 3 4 ⎤
10 –5 2 ⎦
4 2 –3
−2 7 –3
It is probably prey clear why we call those entries “the diagonal.” Here is the
formal definion.
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