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review_of_matrices_and vectors - ,3rded MatricesandVectors...

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Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Objective To provide background material in support of topics in Digital Image Processing that are based on matrices and/or vectors. Review Matrices and Vectors
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Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors An m × n (read "m by n") matrix , denoted by A , is a rectangular array of entries or elements (numbers, or symbols representing numbers) enclosed typically by square brackets, where m is the number of rows and n the number of columns. Some Definitions
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Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Definitions (Con’t) A is square if m = n . A is diagonal if all off-diagonal elements are 0, and not all diagonal elements are 0. A is the identity matrix ( I ) if it is diagonal and all diagonal elements are 1. A is the zero or null matrix ( 0 ) if all its elements are 0. The trace of A equals the sum of the elements along its main diagonal. Two matrices A and B are equal iff the have the same number of rows and columns, and a ij = b ij .
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Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Definitions (Con’t) The transpose A T of an m × n matrix A is an n × m matrix obtained by interchanging the rows and columns of A . A square matrix for which A T = A is said to be symmetric . Any matrix X for which XA = I and AX = I is called the inverse of A . Let c be a real or complex number (called a scalar ). The scalar multiple of c and matrix A , denoted c A , is obtained by multiplying every elements of A by c . If c = - 1, the scalar multiple is called the negative of A .
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Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Definitions (Con’t) A column vector is an m × 1 matrix: A row vector is a 1 × n matrix: A column vector can be expressed as a row vector by using the transpose:
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Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Some Basic Matrix Operations The sum of two matrices A and B (of equal dimension), denoted A + B , is the matrix with elements a ij + b ij . • The difference of two matrices, A - B , has elements a ij - b ij . The product , AB , of m × n matrix A and p × q matrix B , is an m × q matrix C whose ( i , j )-th element is formed by multiplying the entries across the i th row of A times the entries down the j th column of B ; that is,
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Digital Image Processing, 3rd ed.
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