Page 139 - 35Linear Algebra

7.3 Properties of Matrices 139

The r × r diagonal matrix with all diagonal entries equal to 1 is called
the identity matrix, I r , or just I. An identity matrix looks like

 
1 0 0 · · · 0
 0 1 0 · · · 0 
 
 0 0 1 · · · 0  .

I = 
. . . . .
 . . . . . . . . . . 
0 0 0 · · · 1
The identity matrix is special because

I r M = MI k = M

for all M of size r × k.

i
Definition The transpose of an r ×k matrix M = (m ) is the k ×r matrix
j
T
i
M = ( ˆm )
j
j
i
with entries that satisfy ˆm = m .
j
i
T
A matrix M is symmetric if M = M .
Example 86
 
T 2 1
2 5 6
=  5 3  ,
1 3 4
6 4
and
T

2 5 6 2 5 6 65 43
= ,
1 3 4 1 3 4 43 26
is symmetric.

Reading homework: problem 3

Observations

• Only square matrices can be symmetric.

• The transpose of a column vector is a row vector, and vice-versa.

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