Solving Matrix Equations with Real Scalars

In summary, the diagonal entries in A(rB) are unaffected by transposition, and so (A(rB))_{ij}=\sum_k A_{ik}(rB)_{kj}. The equality (AB)_{ij}=\sum_k A_{ik}B_{kj} is also true, since (AB)_{ij}=\sum_k A_{ik}B_{kj}. Finally, Tr(ATA)>0.
  • #1
blueberryfive
36
0
Hello,

A(rB) = r(AB) =(rA)B where r is a real scalar and A and B are appropriately sized matrices.

How to even start? A(rbij)=A(rB), but then you can't reassociate...

Also, a formal proof for Tr(AT)=Tr(A)?

It doesn't seem like enough to say the diagonal entries are unaffected by transposition..

Lastly, let A be an mxn matrix with a column consisting entirely of zeros. Show that if B is an nxp matrix, then AB has a row of zeros.

I can't figure out how to make a proof of this. I know how to say what such and such entry of AB is, but I don't know how to designate an entire column. How do you formally say it will be equal to zero, then...just because the dot product of a zero vector with anything is 0?
 
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  • #2
blueberryfive said:
Hello,

A(rB) = r(AB) =(rA)B where r is a real scalar and A and B are appropriately sized matrices.

How to even start? A(rbij)=A(rB), but then you can't reassociate...
You don't have to. Every entry in "rB" has a factor of r so every entry in A(rB) has a factor of r so A(rb)= r(AB)= (rA)B

Also, a formal proof for Tr(AT)=Tr(A)?

It doesn't seem like enough to say the diagonal entries are unaffected by transposition..
Why not? Would it be better to say "[itex]A^*_{ij}= A_{ji}[/itex]" so that, replacing j with i, "[itex]A^*_{ii}= A_{ii}[/itex]"? That may look more "formal" but it is really just saying that "the diagonal entries are unaffected by transposition".

Lastly, let A be an mxn matrix with a column consisting entirely of zeros. Show that if B is an nxp matrix, then AB has a row of zeros.
[tex](AB)_{ij}= \sum A_{ik}B_{kj}[/itex]. If the "jth" column of B is all 0s, then the "jth" row of A is all 0.

I can't figure out how to make a proof of this. I know how to say what such and such entry of AB is, but I don't know how to designate an entire column. How do you formally say it will be equal to zero, then...just because the dot product of a zero vector with anything is 0?
 
  • #3
Thank you.

Also,

Tr(ATA)[itex]\geq[/itex]0.

I can't even see how to begin...
 
  • #4
blueberryfive said:
Thank you.

Also,

Tr(ATA)[itex]\geq[/itex]0.

I can't even see how to begin...
That's a sum of squares!
 
  • #5
blueberryfive said:
Hello,

A(rB) = r(AB) =(rA)B where r is a real scalar and A and B are appropriately sized matrices.

How to even start? A(rbij)=A(rB), but then you can't reassociate...
The definition of rA where r is a real number and A is a matrix is [itex](rA)_{ij}=rA_{ij}[/itex]. The definition of AB where both A and B are matrices is [itex](AB)_{ij}=\sum_k A_{ik}B_{kj}[/itex]. It's not hard to use these definitions to show that the equalities you mentioned are true. Start with [itex](A(rB))_{ij}=\sum_k A_{ik}(rB)_{kj}[/itex].

All your other questions are also quite easy to answer if you just use these definitions, and the definition of the trace and the transpose: [itex]\operatorname{Tr}A=\sum_i A_{ii},\quad (A^T)_{ij}=A_{ji}[/itex].
 
Last edited:

What is a matrix equation?

A matrix equation is an equation in which a matrix (a rectangular array of numbers) is multiplied by a vector (a column of numbers) to produce another vector.

What are real scalars?

Real scalars are numbers that are not part of a matrix, but are used to multiply or divide the matrix.

What is the process for solving a matrix equation with real scalars?

The process for solving a matrix equation with real scalars involves using the properties of matrix multiplication to simplify the equation, isolating the variable by performing inverse operations, and then solving for the variable using basic algebra.

What are some common mistakes when solving matrix equations with real scalars?

Some common mistakes when solving matrix equations with real scalars include forgetting to use the properties of matrix multiplication, not isolating the variable correctly, and making errors in basic algebra calculations.

How is solving matrix equations with real scalars useful in the field of science?

Solving matrix equations with real scalars is useful in science because it allows for the representation and manipulation of data in a concise and efficient manner. It is commonly used in fields such as physics, engineering, and statistics to solve complex systems of equations and analyze data sets.

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