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Section 1.4
Solid Mechanics Part III
Kelly
22
1.4 Matrices and Element Form
1.4.1
Matrix – Matrix Multiplication
In the next section, §1.5, regarding vector transformation equations, it will be necessary
to multiply various matrices with each other (of sizes
1
3
×
,
3
1
×
and
3
3
×
).
It will be
helpful to write these matrix multiplications in a shorthand element form.
First, it has been seen that the dot product of two vectors can be represented by
[ ][ ]
v
u
T
, or
i
i
v
u
.
Similarly, the matrix multiplication
[ ][ ]
T
v
u
gives a
3
3
×
matrix with element form
j
i
v
u
or, in full,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
v
u
v
u
v
u
v
u
v
u
v
u
v
u
v
u
v
u
This type of matrix represents the
tensor product
of two vectors, written in symbolic
notation as
v
u
⊗
(or simply
uv
).
Tensor products will be discussed in detail in a later
section.
Next, the matrix multiplication
[]
[]
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
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 Fall '11
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