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matrixREV - 2 the columns are linearly independent which...

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MATH 580: Review of Matrices Fall 2004 AB, Penn State Given B , a m × n (rows by columns) matrix: 1. B T B is square and symmetric 2. if B has linearly independent columns (i.e. B has rank n , assuming n < m ) then B T B is symmetric positive definite, which means that ( B T B ) - 1 exists. For a square matrix, A n × n , any of the following is necessary and sufficient for A to be nonsingular: (thus any one implies the others) 1. the columns a of A span R n , i.e. a
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Unformatted text preview: 2. the columns are linearly independent, which means that Ax = 0 has only the solution x = 0, i.e. the nullspace N ( A ) = { } 3. the rows of A span R n 4. the rows are linearly independent 5. det A 6 = 0 6. the inverse A-1 exists, such that A-1 A = AA-1 = I 7. For Ax = λx , λ 6 = 0 (there are no zero eigenvalues of A ) 8. A T A is symmetric positive definite...
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