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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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APPENDIX A. A primer on linear matrix algebra
Note: this is a simplified presentation for finite-dimensional real vector spaces. For more general results and rigorous mathematical definitions, refer to mathematical textbooks.
Matrix. A matrix
of dimension
is a two-dimensional array of real coefficients 
where 
is the line index, 
is the column index. A matrix is usually represented as a table:
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A matrix for which
is called a square matrix.Diagonal. The diagonal of a square
matrix 
is the set of 
coefficients 
. A matrix is called diagonal if
all its non-diagonal coefficients are zero.Transpose. The transpose of a
matrix 
is a 
matrix
denoted 
with the coefficients defined by 
i.e. the coefficients 
and 
are swapped, which looks like a symmetry with respect
to the diagonal:
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Symmetry. A square matrix is symmetric if it is equal to its transpose, i.e.
. This is equivalent
to having 
for any 
and 
. A property of diagonal matrices is that
they are symmetric.Scalar multiplication. A
matrix 
times a real scalar 
is defined as the 
matrix 
with coefficients
.Matrix sum. The sum of two
matrices 
and 
is defined
as the 
matrix 
with coefficients
. It is easy to see that the sum and scalar
multiplication define a vector space structure on the set of 
matrices (the sum is associative and its neutral element
is the zero matrix, with all coefficients set to zero).Matrix product. The product between an
matrix 
and a
matrix 
is defined
as the 
matrix 
with coefficients
given by
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The product is not defined if the number of columns in
is not the same as the number of lines in 
. The product is not commutative in general. The neutral
element of the product is the identity matrix 
defined as the diagonal matrix with values 1 on the diagonal, and
the suitable dimension. If 
the product can
be generalized to matrix times vector 
by identifying
the right-hand term of the product with the column 
of vector coordinates in a suitable basis; then the multiplication
(on the left) of a vector 
by a matrix
can be identified to a linear application from
to 
. Likewise,
matrices can be identified with scalars.Matrix inverse. A square
matrix 
is called invertible
if there exist an 
matrix
denoted 
and called inverse of 
, such
that 
Trace. The trace of a square
matrix 
is defined as the scalar 
which is the sum of the diagonal coefficients.Useful properties.
(A, B, C are assumed to be such that the operations below have a meaning)
The transposition is linear:
Transpose of a product:
Inverse of a product:
Inverse of a transpose:
Associativity of the product:
Diagonal matrices: their products and inverses are diagonal, with coefficients given respectively by the products and inverses of the diagonals of the operands.
Symmetric matrices: the symmetry is conserved by scalar multiplication, sum and inversion, but not by the product (in general).
The trace is linear:
Trace of a transpose:
Trace of a product:
Trace and basis change:
, i.e. the trace is an intrinsic property of the linear application
represented by 
.Positive definite matrices. A symmetric matrix
is defined to be positive definite if, for
any vector 
, the scalar 
unless 
. Positive definite matrices have real positive
eigenvalues, and their positive definiteness is conserved through inversion.Training Course Notes Front Page >>
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03.12.2001 |
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