Download 2.2

MAT 2401
Linear Algebra
2.2 Properties of Matrix
Operations
http://myhome.spu.edu/lauw
Today
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Written HW
Review
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We have defined the following matrix
operations
“term-by-term” operations
• Matrix Addition and Subtraction
• Scalar Multiplication
Non-“term-by-term” operations
• Matrix Multiplication
Review
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We have studied some of the
properties such as…
• AI=IA=A
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In general,
• AB≠BA
• AB=0 does not imply A=0 or B=0
Preview
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Look at more properties about these
operations.
Most of the properties are natural to
conceive (inherited from the number
system).
Sometimes, it may be more effective
to remember what properties are not
true.
Preview
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Most properties come with names. We
will not emphasize on them.
Look at another operation: Transpose
Matrix Addition and
Scalar Multiplication
Let A,B,C be mxn matrices, 0 the mxn zero
matrix, and c and d scalars.
1. A + B = B + A
2. (A + B) + C = A + (B + C)
3. c(dA) = (cd)A
4. c(A + B) = cA + cB
5. (c + d)A = cA + dA
Matrix Addition and
Scalar Multiplication
Let A,B,C be mxn matrices, 0 the mxn zero
matrix, and c and d scalars.
6. A + 0 = A
7. A + (-A) = 0
8. If cA=0, then either c=0 or A=0
Example 1
Solve the matrix equation 3X+A=B
where A  1 0 , B   0 1
1
2
 1 2
Matrix Multiplication
Let A,B,C be matrices of the appropriate
sizes, I a suitably sized identity matrix,
and c and d scalars.
1. (AB)C = A(BC)
2. A(B+C)=AB+AC
3. (A+B)C = AC+BC
4. c(AB)=(cA)B=A(cB)
Cancellation Law
Q: Does AC=BC imply A=B?
A:
Matrix Power
Let A be a square matrix, k a nonnegative integer.
I
if k  0

k
A   A  A   A if k  0
 k times
Laws of Exponents
Let A be a square matrix, i, j, k nonnegative integers.
1. AiAj =
2. (Ai)j =
3. 0k =
4. Ik =
Transpose of a Matrix
Let A=[aij] be a mxn matrix, the
transpose of A is the nxm matrix AT so
that the (i,j)th entry of AT is aji.
(Interchanging the rows and columns of
A)
Transpose of a Matrix
Let A=[aij] be a mxn matrix, the
transpose of A is the nxm matrix AT so
that the (i,j)th entry of AT is aji.
 a11 a12
A   a21 a22
 a31 a32
a13
a23
a33
a14 
a24 
a34 



T
A 










Example 2
 1 2
A


1
0


A 
T
1 2 3 1
T
B
B


 2 2 3 1
C  [x
y
z]
CT 
Scratch:
Q: What is the dimension of the
transpose?
Properties of Matrix Transpose
Let A,B be matrices of the appropriate
sizes, and c a scalar.
1. (AT)T= A
2. (A + B)T = AT + BT
3. (cA)T = cAT
4. (AB)T = BTAT
Properties of Matrix Transpose
Let A,B be matrices of the appropriate
sizes, and c a scalar.
1. (AT)T= A
2. (A + B)T = AT + BT
3. (cA)T = cAT
4. (AB)T = BTAT Why?
Example 3
 1 0 
 0 1
A
,
B


 1 2 
 1 2


AB 
 AB 
T

BT AT 
Symmetric Matrix
A square matrix is symmetric if aij=aji
for all i,j.














Symmetric
Properties of Symmetric
Matrices
1. If A is symmetric, then AT =
In fact,
A is symmetric if and only if AT =
2. AAT and ATA are symmetric for any
matrix A.
Properties of Symmetric
Matrices
1. If A is symmetric, then AT =
In fact,
A is symmetric if and only if AT =
2. AAT and ATA are symmetric for any
matrix A..
Why?