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notes-matrices

# notes-matrices - Notes on matrices and systems Patrick De...

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Unformatted text preview: Notes on matrices and systems Patrick De Leenheer * January 13, 2009 1 Matrices Matrices are tables of generally complex numbers. Examples: A = 1 2 3 4 ,B = 1 2 3 4 5 6 Matrix A is said to be square because it has equally many rows and columns, namely 2. Matrix B on the other hand is rectangular and has 2 rows and 3 columns. A general matrix X may have n rows and m columns, and we sometimes write X ∈ R n × m if all entries of the matrix are real numbers, or X ∈ C n × m if they are complex. Special matrices are row vectors ( n = 1) and column vectors ( m = 1). Operations on matrices We can add matrices X and Y if (and only if) both have the same number of rows and columns, and this is done entrywise: 1 2 2 1 + 1 3 2 0 = 2 5 4 1 We cannot add the matrices A and B defined earlier. We can multiply matrix X and Y (if and only if) matrix X has the same number of columns as the matrix Y has rows, i.e. if and only if X ∈ C n × m and Y ∈ C m × p . The result of the multiplication is a matrix Z = XY ∈ C n × p , i.e. it has the same number of rows as X and the same number of columns as Y . So what is Z ? Let’s specify each of its entries. Denoting [ Z ] ij as the entry in the i th row and j th column of Z , we have that [ Z ] ij = m X k =1 [ X ] ik [ Y ] kj , for i = 1 ,...,n and j = 1 ,...,p. Notice that this formula is also given by the dot product of the i th row of X and the j th column of Y . For example, we can calculate AB (but not BA; why not?) and AA which we denote for short as A 2 : AB = 9 12 15 19 26 33 ,A 2 = 7 10 15 22 We can multiply a matrix by a scalar : If X ∈ C n × m and α ∈ C then [ αX ] ij = α [ X ] ij . For example, 2 A = 2 4 6 8 . The following rules are easily verified, provided that the operation makes sense. The matrix 0 is a matrix having only zero entries, and the matrix I n is a matrix in R n × n with all diagonal entries equal to 1, and off-diagonal entries equal to 0: [ I n ] ii = 1 for all i = 1 ,...,n and [ I n ] ij = 0 * Email: [email protected] Department of Mathematics, University of Florida. 1 if i 6 = j . ( A + B ) + C = A + ( B + C ) A + 0 = A = 0 + A A + B = B + A A + (- A ) = 0 = (- A ) + A ( AB ) C = A ( BC ) I n A = A = AI m A ( B + C ) = AB + AC AB and BA are not necessarily the same, even if both exist (can you give an example?). Determinants : Given a square matrix X ∈ C n × n , we can associate a complex number to it, denoted as det A . I am not giving the general definition which requires some linear algebra , but here is how you compute the determinant for matrices in C 2 × 2 and C 3 × 3 : det a b c d = ad- bc, det x 11 x 12 x 13 x 21 x 22 x 23 x 31 x 32 x 33 = x 11 x 22 x 33 + x 12 x 23 x 31 + x 13 x 32 x 21- x 13 x 31 x 22- x 21 x 12 x 33- x 11 x 23 x 32 We say that X is singular if and only if det X = 0, and non-singular otherwise....
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