multiplying matrices

Nov 2009
75
0
hi!

i have a simple linear algebra question,which will help me understand a previous problem i posted better.
can anyone verify that the following is right?

if i have 3 matrices defined by, A and C where A' = transpose of A
is the following right...?

ACA' = A(ACA')A = (A^2)*C

thanks
i appreciate your help!!
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
hi!

i have a simple linear algebra question,which will help me understand a previous problem i posted better.
can anyone verify that the following is right?

if i have 3 matrices defined by, A and C where A' = transpose of A
is the following right...?

ACA' = A(ACA')A = (A^2)*C

thanks
i appreciate your help!!
What are you trying to accomplish?
 
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Nov 2009
75
0
i just want to know if as a general rule this works, i.e. for all cases?

or would it matter if the problem i was solving was different?

so i can establish this in my head
it makes sense because multiplication is commutative . i could be wrong as usual

thanks
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
i just want to know if as a general rule this works, i.e. for all cases?

or would it matter if the problem i was solving was different?

so i can establish this in my head
it makes sense because multiplication is commutative . i could be wrong as usual

thanks
Matrix multiplication isn't commutative.
 
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dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Example:

\(\displaystyle \begin{bmatrix}
a & b\\
c & d\\
e & f
\end{bmatrix}\begin{bmatrix}
a\\
b
\end{bmatrix}\)

This is defined but if you switch the order the multiplication isn't defined
 
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Nov 2009
75
0
so for example
as in my recent post, if i had

P(A) = ACA' and want to obtain ∂P/∂A

where A,C are matrices

would this be correct for me to do?

ACA' = A(ACA')A = (A^2)*C

so...


∂P/∂a = 2aC


is this correct and does ACA' = A(ACA')A = (A^2)*C work for all cases in any problem like the one i just did?
 
Nov 2009
75
0
Example:

\(\displaystyle \begin{bmatrix}
a & b\\
c & d\\
e & f
\end{bmatrix}\begin{bmatrix}
a\\
b
\end{bmatrix}\)

This is defined but if you switch the order the multiplication isn't defined

thanks for reminding me . of course!!
i was thinking n by n matrices

not all matrices are commutative, but n by n ones are ( ?? ) (Thinking)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
hi!

i have a simple linear algebra question,which will help me understand a previous problem i posted better.
can anyone verify that the following is right?

if i have 3 matrices defined by, A and C where A' = transpose of A
is the following right...?

ACA' = A(ACA')A = (A^2)*C

thanks
i appreciate your help!!
In general ABA is NOT equal to B! And A(ACA)A'= (A^2)C(AA') which is equal to (A^2)C only if AA'= I which is not always true.
 
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