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# Algebraic Properties of Matrix Operation

Recall the Properties of Real Numbers:
• Match the following

 Commutative for multiplication Associative for multiplication 3 (5 + 4) = 3(5) + 3(4) Multiplicative identity Distributive 5+(–5)=0 Identity for multiplication 5 + 4 = 4 + 5 Commutative for addition Associative for addition 5 ( 1/5) = 1 Identity for addition 5 + 0 = 5 Additive identity (5 + 4) + 3 =5 + (4 + 3)

• Of the above, which do you think hold true for matrices?
Commutative for multiplication
Associative for multiplication
Multiplicative identity
Distributive
Identity for multiplication

Properties of Matrix Operations:
Theorem 7: The following are true for matrices (recall that size of the matrices are important):
A + B = B + A
(A + B) + C = A + (B + C)
There exists a unique matrix O (called the zero matrix) such that A + O = A
There exists a unique matrix P such that A + P = O

• We have already seen that AB = BA is NOT true for matrices

• Theorem 8: The following are also true for matrices (recall that size of the matrices are important)
A(BC) = (AB)C
For scalars r and s, r(sA) = (rs)A = s(rA)
r(AB) = (rA)B = A(rB)

• Theorem 9: The following are also true for matrices (recall that size of the matrices are important)
(A + B)C = AC + BC
A(B + C) = AB + AC
(r + s)A = rA + sA
r(A + B) = rA + rB
• Page 69, Exercises 2 and 4

Transpose of a Matrix:
• For any element aij in a matrix A, the transpose, denoted AT = (bij), where bij = aji for all i,j

• For example, for the matrix ,

• Theorem 10:
(A + B)T = AT + BT
(AC^)T = CT AT
(AT)T = A

• A matrix A is symmetric if A = AT

• An nxn matrix is called a square matrix

• In a matrix, the values aii are called the main diagonal
• Page 69, Exercise 8

The Identity Matrix:
• The nxn identity matrix, denoted as In is the square matrix with 1’s on the main diagonal and 0’s everywhere else.

• For example,

• The identity matrix is the multiplicative identity for matrix multiplication. This means that
A In = A = In A

Scalar Products and Vector Norms:
• A vector is an nx1 matrix, usually denoted with and .
NOTE: the book uses bold notation for vectors.

• The scalar product for two vectors is simply matrix multiplication

• In a similar fashion,

• The norm is given by
• For two vectors, the Euclidean distance is given by
• Page 69, Exercises 14 and 20