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Unformatted text preview: B = 6
4 −1
7 AB ≠ BA Calculate first the product AB: 1× 0 + 2 × 6 1× − 1 + 2 × 7 12 13 AB = = 24 25
3 × 0 + 4 × 6 3 × − 1 + 4 × 7 Now reverse the order of operands: 0 ×1 −1×3
BA = 6 ×1 + 7 ×3 0 × 2 −1× 4 − 3 = 27
6 × 2 + 7 × 4 − 4
40 22 Identity Matrices
Definition: Identity matrix is a square matrix with 1s in its principal
Identity
square
diagonal and 0s everywhere else.
diagonal
Notation: I n or I where n is the matrix’ dimension.
1
I = I 3 = 0 0 Properties: 0
1
0 IA =AI =A
I k =I 0
0 1 Watch out for the dimension of I!
1
IA = 0 01 2 3 1 2 3 =
=A
1 2 0 3 2 0 3 1 0 0
1 2 3 1 0 3 AI = 0 1 0 = 2 0 3 = A
2 0 3 0 0 1 23 Null Matrices
Definition: A null matrix (zero matrix) is a matrix of arbitrary dimension
of
all of whose elements are zeroes.
all
Notation: 0 Properties: 0
0= 0 = 2x2
0 0
0 0 0 0
0 = 0 = 0 0 0 3x3
0 0 0 A+0 = 0+ A = A A× 0 = 0× A = 0 24 Idempotent Matrices
Definition: A matrix is idempotent if its product with itself remains itself. AA = A
Obviously, an identity matrix is idempotent:
Obviously, An example of an
An
idempotent matrix:
idempotent 1
0 0 0
1/ 2
1/ 2 II = I 0
1 / 2 1 / 2 25 Idiosyncrasies of Matrix Algebra
Identity matrices are similar to the number 1 in scalar algebra:
Identity
multiplication by an identity matrix does not change the matrix (however,
watch out for dimensions!)
watch
Null matrices are similar to zeroes in scalar algebra. Again, dimensions
Null
do matter.
do
Very importantly, the commutative law for multiplication does not
Very
commutative
hold in case of the matrix algebra: BA
AB ≠
Also, the product of two nonzero matrices is not necessarily zero (as
Also,
two
A = 0
it would be the case with scalars):
it 2 4 − 2 4 AB = 0 ⇒ /
AB = =0
B = 0
1 2 1 − 2 Another idiosyncrasy: CD = CE ⇒D = E
/ 2
C =
6 3
1 D = 9
1 CD = CE , D ≠ E 1
−2
E = 3
2 1
2 26 Transposes
Remember the distinction between a row vector and a column vector: Column vector: x1 x= x 2 Row vector: x' = ( x1 x2 ) We can say that one can be obtained from another by interchanging
We
columns with rows.
columns () Definition: Matrix A' AT is called a transpose of matrix A if it is '
transpose
obtained from the latter by interchanging columns and rows, i.e. if aij 3 8 9
A= 1 0 4 3 1 A' = 8 0 9 4 = a ji Transposes have a “reverse” dimension compared to the original matrix: if A
Transposes
is dimension 2x3, its transpose is dimension 3x2. Square matrices do not
change their dimension when transposes are taken.
change 27 Symmetric Matrices
Definition: A square matrix that is equal to its transpose is called a
symmetric matrix.
symmetric
An example of a symmetric matrix: 1 0 4
D = 0 3 7 4 7 2 It is easy to observe that in matrix D all elements are the symmetric
It
with respect to the principal diagonal: d ij = d ji
with 28 Properties of Transposes
The obvious property: The easy property: ( A') ' = A ( A + B )′ = A'+B' ( AB )′ = B′A′
The dimensionality argument: suppose A , B
The
The reverse order prope...
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 Fall '11
 Kim

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