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Matrix Operations and Definitions: A Comprehensive Overview

This introduction covers matrix definitions, properties, and basic operations including addition, multiplication, powers, transposes, and Boolean matrices. It explains key concepts with examples and outlines the properties of square matrices. Learn about symmetric and Zero-One matrices.

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Matrix Operations and Definitions: A Comprehensive Overview

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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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