Maths(二)---Matrix Arithmetic and Relations(1)

matrix

　7 2 7

　｛6 3 5｝

　　1 2 2　　is a 3 ×3 matrix

Matrix Notation

｛Ａ１１　Ａ１２｝

　Ｂ２１　Ｂ２２

　Ｃ３１　Ｃ３２

Addition and Subtraction of Matrices

例一.

{7 9 5}+{427 429 427} = {434 438 432}

{3 6 3 } 　{ 369 371 369} 　　{ 372 377 372}

例二.

{427 429 427} - {7 9 5}= {420 420 422}

{369 371 369 }　{3 6 3 }　{366 365 366 }

Multiplication of Matrices

例一.

4 {7 9 5 }={28 36 20}

　{3 6 3} 　{12 24 12}

Multiplication of Two Matrices

例一.

{1 5}　　{8 4 3 1}=　　　{18 29 43 31}

{2 7} 　　{2 5 8 6 }　　　{30 43 62 44}

{3 4 }　　　　　　　　　{32 32 41 27}

NOTE:

In general, the product of an (c × d) matrix and an (d × e) matrix has order (c × e).

＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊

例一.

Consider if we were to write an algorithm for matrix multiplication to multiple two(m × m) matrices;

(a) How many steps of addition and multiplication of integers are used?

(b) Given matrix multiplication is associative [i.e. (B1B2)B3 = B1(B2B3)], would suchpreference in the order of multiplication affects the number of steps taken?

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(a) Multiplying two (m × m) matrices C and D, there would be m2 entries in theproduct.

Total numbers of multiplication = m3

Total numbers of addition = m2(m - 1)

(b) If B1 is a (30 × 20) matrix, B2 is a (20 × 40) matrix, and B3 is a (40 × 10) matrix,then steps involved for (B1B2)B3:

Total numbers of multiplication = 30 × 20 × 40 + 30 × 40 ×10 = 36000.

and steps involved for B1(B2B3):

Total numbers of multiplication = 20 × 40 × 10 + 30 × 20 × 10 = 14000.

Clearly, we can see that the second method is much more efficient.

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Transpose of a Matrix

例一

M = {4 6} 　　　　　　　　　　{4 8 5}

　　{8 3} then MT= 　　　　　　{6 3 1}

　　{5 1},

Special Matrices

A square matrix M is symmetric if M = MT

M = {1 2 5} 　　　　　{1 2 5}

{2 8 9} and MT= 　　　{2 8 9}

{5 9 4} 　　　　　　　{5 9 4}

A diagonal matrix M is a square matrix with all elements 0 except those on the leading diagonal

M={1 0 0}

{0 8 0}

{0 0 4}

A unit matrix I is a diagonal square matrix with all elements 0 except those on the leading diagonal and elements of leading diagonal are 1.

I= {1 0 0}

{0 1 0}

{0 0 1}

I M = M I = M

A null matrix N is a matrix with all elements zero.

N= {0 0 0}

{0 0 0}

{0 0 0}