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ACMGM015

1.5 Binary and Permutation Matrices

Applications of Binary Matrices

Note: if you cannot remember what a binary matrix is, revise notes for 1.1 Matrices: Definition and Unique Cases.

  • Binary matrices are used to represent correlation between datasets. If two categories are correlated, a 1 is placed in the element corresponding to the two, if they are not, a 0 is placed instead.
  • They are also used to represent dominance. If one category is “greater than” another, a 1 is placed in the corresponding element, else a 0 is placed instead.

Note: we will further explore dominance matrices in notes 1.7 Matrix Applications: Dominance Matrices.

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1.2 Elementary Matrix Operations

Addition and Subtraction

  • Matrix addition and subtraction is carried out on an element-by-element basis. The element C_{i,j} in the resultant matrix is found by carrying out addition or subtraction with the corresponding elements; A_{i,j} and B_{i,j} in the other matrices.
  • Matrices can only be added or subtracted from one another if they have the same dimensions.

Examples

\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]+\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 2 & 2 \\ 1 & 1 \end{array}\right]

\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]-\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{cc} 0 & 2 \\ -1 & 1 \end{array}\right]

Scalar Multiplication

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