Transcription of MATRIX ALGEBRA REVIEW - University of Nevada, Reno
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MATRIX ALGEBRA REVIEW - University of Nevada, Reno
MATRIX ALGEBRA REVIEW (PRELIMINARIESA MATRIX is a way of organizing is a rectangular array of elements arranged in rows and columns. For example, the following matrixA has m rows and n elements can be identified by a typical element ija, where i=1,2,..,m denotes rows and j=1,2,..,ndenotes MATRIX is of order (or dimension) m by n (also denoted as (m x n)).A MATRIX that has a single column is called a column MATRIX that has a single row is called a row transpose of a MATRIX or vector is formed by interchanging the rows and the columns. A MATRIX oforder (m x n) becomes of order (n x m) when example, if a (2 x 3) MATRIX is defined by Then the transpose of A, denoted by A , is now (3 x 2) AA= )( AkkA = )(, where k is a scalar. = =232221131211aaaaaaA = 231322122111aaaaaaA2 SYMMETRIC MATRIXWhen AA= , the MATRIX is called symmetric. That is, a symmetric MATRIX is a square MATRIX , in that ithas the same number of rows as it has columns, and the off-diagonal elements are symmetric ( ).)
6 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 1 2 9 5 8 10 7 28 3 9 6 21 4 5 2 14 A has rank 3 because A = 0, but 63 0 8 10 7 3 9 6 4 5 2 = ≠ That is, the largest submatrix of A whose determinant is not zero is of order 3.
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