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More Linear Algebra Fun!
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27-11-2010 8:39pm#1I'm so angry :mad: You know basic matrix multiplication? Every book I've
looked in, & I've spent all frickin' day on googlebooks & amazon checking
this out, defines matrix multiplication in the shorthand summation notation
or else as the standard row-column algorithm you should parrot off.
Every forum I've read defines matrix multiplication as these things -
I mean it's definition, who am I to question it? - or else says
'it's because it works, it's just a convenient way to define it this way'
or else uses some linear transformation explanation that I haven't studied
yet (but will in 1 chapter!!) but this linear transformation thing doesn't look
convincing to me from what I understand of it. Basically the only person
who went to the trouble of explaining this properly was Sal of
khanacademy :cool: here Did you know about this (I'll explain in a minute)?
Where &/or in what book did you learn about it?
Well, not every book defines it as these things. My book by Lang gives
a slightly better explanation in terms of dot products but it wasn't
satisfying enough. It was enough of a hint at the right way to do this
but he didn't explain it properly unfortunately.
Basically I am partly posting this to find out more about a specific operation
known as the transpose. I'm actually a little confused because in one
book, page 4, he defines a vector in two ways:
X = (x,y,z)
or
___|x|
X= |y|
___|z|
(Obviously the ___ are just to get the | | things to form a column shape
)
which are equivalent but then in the video I linked to above Sal calls
the vector:
___|x|
X= |y|
___|z|
as if it's normal but calls X = (x,y,z) the transpose, Xᵀ = (x,y,z).
I remember from my old study of linear algebra (which I hated & quit
because it made no sense memorizing algorithms and faking my way through proofs) that
the transpose is different somehow and is used in inverting a matrix
I think but is a vector and it's transpose the same thing or something?
Anyway, using this idea of a transpose it makes the whole concept of
matrix multiplication 100% completely, lovingly, passionately, painfully,
hatefully, relievingly intelligible. I think the picture is clear,
In part 3 you just take the transpose of each row of the 2x3 matrix &
dot it with the 3x1 matrix. I just wrote part 4 in as well because
in that book around page 4 he defines both modes of dot product as
being the same, which they are.
It seems like a trick the way Ive decomposed the matrix though,
I mean I could use these techniques to multiply matrices regardless
of their dimensions.
I've just written down a method using these to multiply two matrices
of dimensions 2x3, i.e. (2x3)•(2x3) and gotten a logical answer.
If we copy the exact algorithm I've used in the picture then multiplying
two matrices of equal size is indeed meaningless as you take the dot
product of two differently sized vectors but if I play with the techniques
used to decompose a big matrix I can swing it so that I get a logical
answer.
[1,2,3][a,b,c] = [1,2,3][a] [1,2,3] [1,2,3][c]
[4,5,6][d,e,g] = [4,5,6][d] [4,5,6][e] [4,5,6][g]
(This is the same as I do in the picture, then instead of transposing
straight away I just use more of this decomposition only this time I
decompose the left matrix instead of the right one):
[1,2,3][a] [1,2,3] [1,2,3][c] = [1][a] [2][a] [1] [2] [3] ...
[4,5,6][d] [4,5,6][e] [4,5,6][g] = [4][d] [5][d] [6][d] [4][e] [5][e] ...
I can view this as a dot product:
[1]•[a] [2]•[a] [1]• [2]• [3]• ...
[4]•[d] [5]•[d] [6]•[d] [4]•[e] [5]•[e] ...
(Obviously I just wrote 2 •'s @ each row vector to keep it neat
)
and I end up with some ridiculously crazy matrix essentially being
meaningless but still following the "rules" I came up with something.
This is important, my little knowledge of Hamilton is that he just defined
ijk = -1 because it worked. Maybe this is true, and from what I know
using this kind of algebra is useful in special relativity but I think it's
literally a cheat, there is no explanation other than "it works".
Hopefully I'm wrong! But, with my crazy matrix up here, why is it wrong?
Is it really just that "it works" when we do it the way described in
the picture but it doesn't work (i.e. it doesn't describe physical reality)
when I do it the way I did here? What does this say about mathematics
being independent from reality when we ignore things like this, my
ridiculous matrix, and focus on the ones that describe reality?
I know it's stupid but I don't knw why :P
I'm also worried because just magically defining these things seems to
be common, looking at differential forms I think, & this is because I
haven't studied them properly, that you literally invoke this witchcraft
when doing algebra with the dx's and dy's.
I seriously hope that there are reasons behind all of this, thinking
about the cross product post I made there was a perfect reason why
things like ixj = k & j x i = -k make sense but here I'm worried.
In the cross product example the use of a determinant, an abuse of
notation, is a clear sign we're invoking magic spells to get the right
answers but with matrix multiplication I haven't even located the
source of the sourcery yet & it's driving me crazy
Honestly, tell me now if I've got more of this to expect with
differential forms or will I get a solid answer??? :pac:
TL;DR - The method in the picture of multiplying matrices seems to me
to be the most logical explanation of matrix multiplication, but why is
it done that particular way & not the way I described in this part:
[1,2,3][a,b,c] = [1,2,3][a] [....
of the post? Also, with differential forms when you multiply differential's
dx's and dy's etc... you are using magic sorcery adding minuses yada
yada yada by definition, how come? Is there a beautiful reason for
all of this like that described in the cross product post? Oh, and what's
the deal with transposes? Transposing vectors is the reason why I can
use this method in the picture, but I mean I could stupidly take the matrix
[1,2,3]
[4,5,6]
as being either:
(1,2,3) transposed from it's column vector form, or
(1,4) & (2,5) & (3,6) as being the vectors, it's so weird...
Also, I could have taken part 2 of the picture differently, multiplying
the Y matrix by 3 1x2 X vectors, again it's so weird....
/pent_up_rant...0
Comments
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I strongly suspect that CJC86 was not relying on your miniscule finishing tag in order to describe your post as a rant.
I came to the same conclusion on the basis of the duck test. 0 -
MathsManiac wrote: »I strongly suspect that CJC86 was not relying on your miniscule finishing tag in order to describe your post as a rant.
I came to the same conclusion on the basis of the duck test.A rant is a speech or text that does not present a calm argument;
rather, it is typically an enthusiastic speech or talk or lecture on an idea,
a person or an institution. Compare with a dialectic.
Rants can be based on partial fact or may be entirely factual but written
in a comedic/satirical form.
Rants can also be used in the defense of an individual, idea or organization.
Rants of this type generally occur after the subject has been attacked by
another individual or group."
http://en.wikipedia.org/wiki/Rant
So asking questions about ideas, giving your reasons for those questions
& highlighting your concerns is nothing but a rant, that's good to know.
If you guys don't know anything about the idea I'm asking about that's
fine but if you're going to try to insult my questions at least insult me
using the right words.0 -
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Sorry sponsoredwalk, I haven't looked through your whole post. It is a bit of an eyeful.
It's probably best to begin by thinking about points in the plane. In that setting, matrices represent very natural geometric transforms: dilations, rotations, projections and reflections off the top of my head, though there may be others.
It's natural to be interested in compositions of transforms. A moment's thought will show you that a rotation followed by a projection is not necessarily the same as that same projection followed by the same rotation: composition of these transforms is noncommutative!
It turns out that all these transforms can be represented by 2x2 matrices, and matrix multiplication is exactly what you need to represent the composition given the two transforms.
Moreover, because of the linearity property, you know exactly what a transform will do to every vector in the plane just by knowing what it does to the vectors (1,0) and (0,1), since any vector (x,y) = x(1,0) + y(0,1).
I know that's not quite what you asked, but for me it's by far the most natural way to think of a linear transformation.0 -
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Holy jesus I've cracked it!!!
I started revising my linear algebra, it's fun - each time you come back
the level of abstraction goes up & with it understanding.
I went back and read some of Arthur Cayley's "A Memoir on the Theory of
Matrices" (.pdf) because I was confronted with the issue of matrix multiplication
again. Needless to say the idea of linear mappings never satisfied me in the
way what I'm about to write does.
If you read Cayley's paper he denotes his matrices as follows:
(a, b, c)
|α, β, γ|
|θ, λ, μ|
He includes the "( )" in the top line of his matrix, why? Well I thought
it was some notational hazard seeing as he's writing in 1858, basically it
just put me on the alert for typo's relating to age.
Then, he wrote something very novel, he tells us that a more convenient
way to represent the system of linear equations:
T = ax + by + cz
U = αx + βy + γz
V = θx + λy + μz
is as follows:
(T,U,V) = (a, b, c)(x,y,z)
________|α, β, γ|
________|θ, λ, μ|
Notice the shape of the (T,U,V) & (x,y,z)! It just looks far more natural
this way. At first I thought it was a notational hazard, (1858!), but then
I thought no because looking at the paper he certainly took advantage
of Gutenberg so I thought about it & jeesh is this notation far clearer!
I want to stress this point because if we write the above system as
a linear combination in the way the notation clearly suggests it is
extremely clear what's going on (first of all) and later on it is pivotal,
so I'll do it:
(T,U,V) = (a, b, c)(x,y,z) = _|a| __ |b| __ |c| = ax + by + cz
________|α, β, γ| _____ = x|α| + y|β| + z|γ| = αx + βy + γz
________|θ, λ, μ| _____ = _|θ| __ |λ| __ |μ| = θx + λy + μz
He goes on to explain that
(T,U,V) = (a, b, c)(x,y,z)
________|α, β, γ|
________|θ, λ, μ|
represents the set of linear functions:
((a, b, c)(x,y,z),(α, β, γ)(x,y,z),(θ, λ, μ)(x,y,z))
I think it's clear that
(T,U,V) = ((a, b, c)(x,y,z),(α, β, γ)(x,y,z),(θ, λ, μ)(x,y,z))
So we see that
T = (a, b, c)(x,y,z) = ax + by + cz
U = (α, β, γ)(x,y,z) = αx + βy + γz
V = (θ, λ, μ)(x,y,z) = θx + λy + μz
If you compare this notation to modern notation:
|T| = |a, b, c||x| = |(ax + by + cz)|
|U| = |α, β, γ||y|=_|(αx + βy + γz)|
|V| = |θ, λ, μ||z| = |(θx + λy + μz)|
multiplication of matrices is extremely clear when multiplying an (A)mxn
matrix by an (X)nx1 matrix. It just follows from unfurling a system.
We're going to use this idea when multiplying general matrices.
He then goes on to define all of the standard abelian operations of
matrices in a nice way in the paper which I really recommend reading but
what comes next is absolutely astonishing.
He defines the standard matrix representation
________
(T,U,V) = (a, b, c)(x,y,z)
________|α, β, γ|
________|θ, λ, μ|
but goes on to define
(x,y,z) = (a', b', c')(ξ,η,ζ)
________|α', β', γ'|
________|θ', λ', μ'|
Just think about all of this so far! No magic or defining strange operations,
matrix decomposition is natural and through it we see that matrix
multiplication is like a kind of linear algebra chain rule, very natural.
It is absolutely brilliant how he just morphed (x,y,z) there!
So:
(T,U,V) = (a, b, c)(x,y,z) = (a, b, c)_(a', b', c')(ξ,η,ζ)
________|α, β, γ| _____ = |α, β, γ| |α', β', γ'|
________|θ, λ, μ| _____ = |θ, λ, μ|_|θ', λ', μ'|
If you read his paper you'll see that he then defines
(a, b, c)_(a', b', c')(ξ,η,ζ) = |A _ B__C |(ξ,η,ζ)
|α, β, γ| |α', β', γ'|_____ = |A'_ B' _C' |
|θ, λ, μ|_|θ', λ', μ'| _____= |A''_B''_ C''|
This is the genius of his idea and derivation, how he gets from the L.H.S.
to the R.H.S. is what I'm going to do in the rest of my post, he skips this
step & it took me quite a while to figure it out, if you're looking for a
challenge read the first few pages of Caley's paper & try to figure it out
without peeking at my solution, it's an exercise :cool:
(NB: This could be extremely easy & I just don't recognise it)
----
(T,U,V) = (a, b, c)(x,y,z) = _|a| __ |b| __ |c|
________|α, β, γ| _____ = x|α| + y|β| + z|γ|
________|θ, λ, μ| _____ = _|θ| __ |λ| __ |μ|
(x,y,z) = (a', b', c')(ξ,η,ζ)
________|α', β', γ'|
________|θ', λ', μ'|
x = a'ξ + b'η + c'ζ
y = α'ξ + β'η + γ'ζ
z = θ'ξ + λ'η + μ'ζ
(T,U,V) = T = _|a| __ |b| __ |c| = _____________|a| + _____________|b| + _____________|c|
________ U = x|α| + y|β| + z|γ| = (a'ξ + b'η + c'ζ)|α| + (α'ξ + β'η + γ'ζ)|β| + (θ'ξ + λ'η + μ'ζ)|γ|
________ V = _|θ| __ |λ| __ |μ| = _____________|θ| + _____________|λ| + _____________|μ|
T = (a'ξ + b'η + c'ζ)•a + (α'ξ + β'η + γ'ζ)•b + (θ'ξ + λ'η + μ'ζ)•c
U = (a'ξ + b'η + c'ζ)•α + (α'ξ + β'η + γ'ζ)•β + (θ'ξ + λ'η + μ'ζ)•γ
V = (a'ξ + b'η + c'ζ)•θ + (α'ξ + β'η + γ'ζ)•λ + (θ'ξ + λ'η + μ'ζ)•μ
T = a'ξ•a + b'η•a + c'ζ•a + α'ξ •b+ β'η•b + γ'ζ•b + θ'ξ•c + λ'η•c + μ'ζ•c
U = a'ξ•α + b'η•α + c'ζ•α + α'ξ•β + β'η•β + γ'ζ•β + θ'ξ•γ + λ'η•γ + μ'ζ•γ
V = a'ξ•θ + b'η•θ + c'ζ•θ + α'ξ•λ + β'η•λ + γ'ζ•λ + θ'ξ•μ + λ'η•μ + μ'ζ•μ
T = (a'a + α'b + θ'c)ξ + (b'a + β'b + λ'c)η + (c'a + γ'b + μ'c)ζ
U = (a'α + α'β + θ'γ)ξ + (b'α + β'β + λ'γ)η + (c'α + γ'β + μ'γ)ζ
V = (a'θ + α'λ + θ'μ)ξ + (b'θ + β'λ + λ'μ)η + (c'θ + γ'λ + μ'μ)ζ
T = |(a'a + α'b + θ'c) (b'a + β'b + λ'c) (c'a + γ'b + μ'c)| (ξ,η,ζ)
U = |(a'α + α'β + θ'γ) (b'α + β'β + λ'γ) (c'α + γ'β + μ'γ)|
V = |(a'θ + α'λ + θ'μ) (b'θ + β'λ + λ'μ) (c'θ + γ'λ + μ'μ)|
(Change a'a to aa' in every entry, makes more sense for what follows)
T = |(aa' + bα' + cθ') (ab' + bβ' + cλ') (ac' + bγ' + cμ')_| (ξ,η,ζ)
U = |(αa' + βα' + γθ') (αb' + ββ' + γλ') (αc' + βγ' + γμ') |
V = |(θa' + λα' + μθ') (θb' + λβ' + μλ') (θc'θ + λγ' + μμ')|
(T,U,V) = |(a,b,c)(a',α',θ') (a,b,c)(b',β',λ') (a,b,c)(c',γ',μ')_| (ξ,η,ζ)
_____ ___|(α,β,γ)(a',α',θ') (α,β,γ)(b',β',λ') (α,β,γ)(c',γ',μ')_|
_____ ___|(θ,λ,μ)(a',α',θ') (θ,λ,μ)(b',β',λ') (θ,λ,μ)(c',γ',μ')_|
There we have it!
Whadda y'all think?
(:D^Me Right Now^:D)
If you multiply out the R.H.S. of:
(a, b, c)(x,y,z) = (a, b, c)_(a', b', c')(ξ,η,ζ)
|α, β, γ| _____ = |α, β, γ| |α', β', γ'|
|θ, λ, μ| _____ = |θ, λ, μ|_|θ', λ', μ'|
in the normal way you do it you see it agrees with the above derivation.
There we have it, a formal justification and it's all self-contained in a
standard Anxn matrix exploiting the unknowns (x,y,z). Obviously this can
be turned into a neverending rabbit hole if we want it to.
----
Edit: I'll add that looking at matrix multiplication this way is an irrefutable explanation of
the reason behind the fact that an (A)mxn matrix by an (X)nxp gives a (B)mxp matrix,
it explains why it doesn't make sense to multiply a matrix of column "n" by a matrix
with rows anything but "n".0 -
I really must say that I'm over the moon since I read Cayley's explanation of
matrices, did any of you check it out? Did you's already know this & just not
understand that this was what I was looking for? If you skipped my post you
should read Cayley's explanation at least, it's just fascinating & I think it
would be a great optional section in a textbook that you could really set
as a theorem to be proven with the mini-calculation I did left as an exercise
mid-proof 0 -
Jesus christ here's another way to multiply matrices :eek:
I got Hoffman/Kunze Linear Algebra 2nd hand in the TCD bookshop the
other day & this book is apparently one of the most rigorous there is at
this level. It's a great book so far and look what fruitful gifts it bears!
Lets re-cap, there is
1) The dot-product formulation,
2) Cayley's clear explanation I've written above,
3) What I'll explain next:
Since two systems of linear equations are equivalent if the equations of one
system are linear combinations of the equations in the other system, we can
use the idea of row-equivalence (the matrix version of system equivalence) to
construct a new matrix whose entries are simply linear combinations of the
(matrix-representative) equations making up a certain system.
So, if B is an mxn matrix we construct a matrix C whose rows are simply
linear combinations of the rows of B.
____|β₁|
____|β₂|
B = _|. |
____|._|
____|._|
____|β₊| (Where the β's are rows)
and
____|γ₁|
____|γ₂|
C =_|._|
____|._|
____|._|
____|γ₊|
The γ's are linear combinations of the rows of B, so
γ₁ = α₁β₁ + α₂β₂ + ... + α₊β₊
γ₂ = α₁β₁ + α₂β₂ + ... + α₊β₊
.
.
.
γ₊ = α₁β₁ + α₂β₂ + ... + α₊β₊
Now comes the conceptual leap, the α's are the column elements of a
different matrix A :eek:
Took me a while to figure this one out but if you take the o'th row:
γ₀ = α₁β₁ + α₂β₂ + ... + α₊β₊
and append the a's with the o as follows:
γ₀ = α₀₁β₁ + α₀₂β₂ + ... + α₀₊β₊
for every row, i.e.
γ₁ = α₁₁β₁ + α₁₂β₂ + ... + α₁₊β₊
γ₂ = α₂₁β₁ + α₂₂β₂ + ... + α₂₊β₊
.
.
.
γ₊ = α₂₁β₁ + α₂₂β₂ + ... + α₂₊β₊
you've got a new way of multiplying matrices giving equivalent results &
it's grounded in some clever theory. Remember the β's are whole rows,
it's crazy! 0 -
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