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MathGroup Archive 2000

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Re: Matrix Multiplication...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23572] Re: [mg23551] Matrix Multiplication...
  • From: BobHanlon at aol.com
  • Date: Sat, 20 May 2000 17:44:29 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

In a message dated 5/20/2000 3:36:50 AM, jrchaff at nwlink.com writes:

>New to Mathematica, and to exotic matrix techniques.
>
>Apparently Mathematica does "ordinary" matrix multiplication
>via the "dot" product symbol, ".";  but when one uses the
>asterisk, one gets "squaring" of two matrices:
>
>If m1 = {{a,b},{c,d}};  and m2={{e,f},{g,h}};  then
>
>m1*m2 = {{ae, bf},{cg,dh}},  and
>
>m1*m1 = {{a^2, b^2},{c^2, d^2}};
>
>while m1.m2 gives normal matrix multiplication.
>
>What is going on here?  What kind of matrix multiplication is
>this "*" giving?  Why does it match up with the casual appearance
>of 'squaring' if m1 = m2?  I tried the Mathematica function
>"Outer", i.e. "Outer[Times,m1,m2]", and this is NOT the same
>as the result with "*".  I don't understand outer products anyway,
>but apparently that is not this, whatever it is.
>
>Can someone give me a short explanation?
>

m1 = {{a, b}, {c, d}};
m2 = {{e, f}, {g, h}};

m1*m2

{{a*e, b*f}, {c*g, d*h}}

(m1*m2) operates in the same manner as (m1+m2) or (m1-m2) or (m1/m2); i.e., 
they all involve pairwise operations on the corresponding elements.

m1 + m2

{{a + e, b + f}, {c + g, d + h}}

m1 - m2

{{a - e, b - f}, {c - g, d - h}}

m1/m2

{{a/e, b/f}, {c/g, d/h}}

These pairwise operations are consistent with the behavior expected for basic 
operations

And @@ Table[Sum[m1, {k, 1, n}] == n*m1, {n, 1, 10}]

True

Sum[m1, {k, 1, n}] == n*m1

True

m1 - m1 == 0*IdentityMatrix[Length[m1]]

True

(m1 + m2) - m2 == m1 + (m2 - m2) == m1

True

And @@ Table[Product[m1, {k, 1, n}] == m1^n, {n, 1, 10}]

True

Product[m1, {k, 1, n}] == m1^n

True

m2*(m1/m2) == (m1*m2)/m2 == m1*(m2/m2) == m1

True

To understand what Outer does, just use a generic function, say f,

Outer[f, m1, m2 ]

{{{{f[a, e], f[a, f]}, {f[a, g], f[a, h]}}, 
  {{f[b, e], f[b, f]}, {f[b, g], f[b, h]}}}, 
 {{{f[c, e], f[c, f]}, {f[c, g], f[c, h]}}, 
  {{f[d, e], f[d, f]}, {f[d, g], f[d, h]}}}}

Which for f replaced by Times is

Outer[Times, m1, m2 ]

{{{{a*e, a*f}, {a*g, a*h}}, {{b*e, b*f}, {b*g, b*h}}}, 
 {{{c*e, c*f}, {c*g, c*h}}, {{d*e, d*f}, {d*g, d*h}}}}

m3 = Outer[Times, m2, m1]/m2;

m3[[1, 1]] == m3[[1, 2]] == m3[[2, 1]] == m3[[2, 2]] == m1

True


Bob

BobHanlon at aol.com


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