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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
∈
R
(
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 A1 exists, such that A1 A = AA1 = 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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This note was uploaded on 07/23/2008 for the course MATH 580 taught by Professor Belmonte during the Fall '04 term at Pennsylvania State University, University Park.
 Fall '04
 BELMONTE
 Applied Mathematics, Matrices

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