mat 2401 linear algebra 2.2 properties of matrix operations
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MAT 2401Linear Algebra
2.2 Properties of Matrix Operations
http://myhome.spu.edu/lauw
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Today
Written HW
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Review
We have defined the following matrix operations
“term-by-term” operations•Matrix Addition and Subtraction
•Scalar Multiplication Non-“term-by-term” operations
•Matrix Multiplication
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Review
We have studied some of the properties such as…•AI=IA=A
In general, •AB≠BA
•AB=0 does not imply A=0 or B=0
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Preview
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.
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Preview
Most properties come with names. We will not emphasize on them.
Look at another operation: Transpose
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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 + A2. (A + B) + C = A + (B + C)3. c(dA) = (cd)A4. c(A + B) = cA + cB5. (c + d)A = cA + dA
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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 = A7. A + (-A) = 08. If cA=0, then either c=0 or A=0
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Example 1
Solve the matrix equation 3X+A=Bwhere 1 0 0 1
, 1 2 1 2
A B
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Matrix Multiplication
Let A,B,C be matrices of the appropriatesizes, I a suitably sized identity matrix, and c and d scalars.
1. (AB)C = A(BC)2. A(B+C)=AB+AC3. (A+B)C = AC+BC4. c(AB)=(cA)B=A(cB)
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Cancellation Law
Q: Does AC=BC imply A=B?A:
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Matrix Power
Let A be a square matrix, k a non-negative integer.
times
if 0
if 0k
k
I kA A A A k
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Laws of Exponents
Let A be a square matrix, i, j, k non-negative integers.
1. AiAj =2. (Ai)j =3. 0k =4. Ik =
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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)
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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.
21 22 23 24
31 32 33 34
11 12 13 14
TA a a a a A
a a a a
a a a a
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Example 2
1 2
1 0
1 2 3 1
2 2 3 1
[ ]
T
T
T
A A
B B
C x y z C
Scratch:Q: What is the dimension of the transpose?
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Properties of Matrix Transpose
Let A,B be matrices of the appropriatesizes, and c a scalar.
1. (AT)T= A2. (A + B)T = AT + BT
3. (cA)T = cAT
4. (AB)T = BTAT
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Properties of Matrix Transpose
Let A,B be matrices of the appropriatesizes, and c a scalar.
1. (AT)T= A2. (A + B)T = AT + BT
3. (cA)T = cAT
4. (AB)T = BTAT Why?
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Example 3
1 0 0 1,
1 2 1 2
T
T T
A B
AB
AB
B A
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Symmetric Matrix
Symmetric
A square matrix is symmetric if aij=aji
for all i,j.
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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.
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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?