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lecture_6- Special Types of Matrices and Partitioned Matrices Part 1

Lecture_6- Special Types of Matrices and Partitioned Matrices Part 1

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MA 265 LECTURE NOTES: FRIDAY, JANUARY 18 Special Types of Matrices In today’s lecture, we only consider n × n matrices. Square Matrices. We give a few definitions for n × n matrices: Let A = [ a ij ] be an n × n matrix i.e. a matrix which has the same number of rows as columns. Such a matrix is called a square matrix . Recall that the elements a 11 , a 22 , ..., a nn compose the main diagonal of A . We say A is upper triangular if a ij = 0 for i > j : A = a 11 a 12 · · · a 1 n 0 a 22 · · · a 2 n . . . . . . . . . . . . 0 0 · · · a nn . Similarly, we say A is lower triangular if a ij = 0 for i < j . Equivalently, A is lower triangular when its transpose A T is upper triangular. We say A is a diagonal matrix if the entries off the main diagonal are zero i.e. a ij = 0 for i negationslash = j . Equivalently, a diagonal matrix is a matrix that is both upper triangular and lower triangular. We say A is a scalar matrix if there is a fixed number r such that a ij = r for i = j yet a ij = 0 for i negationslash = j : A = r 0 · · · 0 0 r · · · 0 . . . . . . . . . . . . 0 0 · · · r . Equivalently, a scalar matrix is a diagonal matrix where all of the elements a kk on the main diagonal are the same number r .
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