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A very brief introduction to

A very brief introduction to. Matrix (Section 2.7). Definitions Some properties Basic matrix operations Zero-One (Boolean) matrices. row. column. Matrix ( Section 2.7).

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A very brief introduction to

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  1. A very brief introduction to Matrix (Section 2.7) • Definitions • Some properties • Basic matrix operations • Zero-One (Boolean) matrices

  2. row column Matrix (Section 2.7) Definition:A matrix is a rectangular array of numbers. A matrix of m rows and n columns is called an mn matrix, denoted Amn. The element or entry at the ith row and jth column is denoted ai,j. The matrix can also be denoted A = [ai,j]. Example a2,3 = 2

  3. Matrix Two matrices Amn and Bpq are equal if they have the same number of rows and columns (m = p and n = q), and their corresponding entries are equal (ai,j =bi,j for all i, j). Amn is a square matrix if m = n, denoted Am A square matrix A is said to be symmetric if ai,j = aj,ifor all i and j.

  4. Matrix arithmetic (operations) Matrix addition. Amn and Bmn • must have the same numbers of rows and columns • add corresponding entries Amn + Bmn = Cmn = [ai,j + bi,j] Matrix subtraction is done similarly

  5. Matrix arithmetic (operations) Multiply a matrix by a number. • bA = [bai,j] (i.e., multiply the number to each entry.) Multiplication of two matrices. Amk and Bkn • number of columns of the first must equal number of rows of the second • the product is a matrix, denoted AB = Cmn • Entry ci,j is the sum of pair-wise products of the ith row of A and jth column of B

  6. Matrix arithmetic (operations) Example

  7. Powers of (square) matrixAn A0 = In =, Ar = AA···A r times Powers and Transposes Identity matrix: In • A square matrix of n rows and n columns • Diagonal entries are 1, all other entries are 0 (ii,i= 1 for all i, ii,j= 0 for all i != j.) • For matrix Amn, we have Im A = A In = A

  8. Powers and Transposes Matrix transpose: Amn • the transpose of A, denoted At, is a n m matrix • At = [bi,j = aj,i] • ith row of A becomes ith column of At Theorem: A square matrixAnis symmetric iff A = At

  9. Zero-One (Boolean) Matrix Definition: • Entries are Boolean values (0 and 1) • Operations are also Boolean • Matrix join. • A  B = [ai,j  bi,j] • Matrix meet. • A  B = [ai,j  bi,j] Example:

  10. Zero-One (Boolean) Matrix • Matrix multiplication: Amk and Bkn • the product is a Zero-One matrix, denoted AB = Cmn • cij = (ai1  b1j)  (ai2  b2i)  …  (aik  bkj). • Example:

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