Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 19
Matriks
The transformation given by the system of equations
is represented as a matrix equation by
where the
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are called matrix elements.
An
set of
denoted
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matrix consists of
rows and
columns, and the
matrices with real coefficients is sometimes
Matrix Addition
Denote the sum of two matrices
and
(of the same
dimensions) by
. The sum is defined by adding
entries with the same indices
over all and . For example,
Matrix addition is therefore both commutative and associative.
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The product
of two matrices
and
is defined as
where
is summed over for all possible values of and
and
the notation above uses the Einstein summation convention. The
implied summation over repeated indices without the presence of
an explicit sum sign is called Einstein summation, and is
commonly used in both matrix and tensor analysis. Therefore, in
order for matrix multiplication to be defined, the dimensions of the
matrices must satisfy
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where
denotes a matrix with
Writing out the product explicitly,
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rows and
columns.
where
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Matrix Equality
Two matrices
and
for all
. Therefore,
while
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are said to be equal iff
Transpose
The object obtained by replacing all
elements
with
. For a second-tensor rank
tensor
, the tensor transpose is simply
. The matrix
transpose, most commonly written
, is the matrix
obtained by exchanging A 's rows and columns, and
satisfies the identity
Bina Nusantara
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Bina Nusantara