A question about the cross product as related to matrix multiplication

[RufusDawes]

I understand that the cross product, in lay mans terms doesn't exist unless we're in 3 dimensions.

When you multiply two matrices together I have been told you get something similar. I hear that this is because a matrix can be treated as a vector.

So if we are talking about measurable things, in the real world does matrix multiplication have any relevance ?

I might be off the mark but what if we were to build a model, based on height weight and income would that be definable in 3 dimensions and therefore if you had two sets of observation and you multiplied them the answer would be nonsense but somehow it has a definition in 3 dimensions as a new matrix ?

if for example, i had a range of multiplies I wanted to apply to each observation or set of observations, stored in a matrix then this would be a scalar multiplication of a matrix onto another matrix ? Does such a thing exist.

Apologies I am not a mathematician so my terminology is buggered but I'm still curious.

Thanks,

 

[SteamKing]

Matrix multiplication is used all the time to design structures, calculate stresses, etc. It's a tool which makes doing certain mathematical calculations easier to organize and handle, especially for computers. It's like asking if regular multiplication has any relevance.

 

[RufusDawes]

Matrix multiplication is used all the time to design structures, calculate stresses, etc. It's a tool which makes doing certain mathematical calculations easier to organize and handle, especially for computers. It's like asking if regular multiplication has any relevance.

okay, so what if I were in grade 1 and didn't understand multiplication ?

Do you have some examples ? Because when I looked at some examples for vector cross product it started to make some sense. Especially when they said it only exists in 3 dimensions.

 

[SteamKing]

The cross product as defined for cartesian coordinates exists and has meaning for 3 dimensions and 7 dimensions.

http://en.wikipedia.org/wiki/Cross_product

The reasons for this are quite mathematically abstract.

The simplest application for matrix multiplication would probably be solving a system of simultaneous linear equations. I can't give you anything simpler than that, just like you would be hard pressed to explain multiplication to a first grader who is just learning to add. Simple matrix manipulations are usually first discussed in algebra classes in middle or high school.

http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php

Simultaneous linear equations:

http://en.wikipedia.org/wiki/System_of_linear_equations

 

[RufusDawes]

The cross product as defined for cartesian coordinates exists and has meaning for 3 dimensions and 7 dimensions.

http://en.wikipedia.org/wiki/Cross_product

The reasons for this are quite mathematically abstract.

The simplest application for matrix multiplication would probably be solving a system of simultaneous linear equations. I can't give you anything simpler than that, just like you would be hard pressed to explain multiplication to a first grader who is just learning to add. Simple matrix manipulations are usually first discussed in algebra classes in middle or high school.

http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php

Simultaneous linear equations:

http://en.wikipedia.org/wiki/System_of_linear_equations

Yes yes but no one ever actually understands what anything means, which is why you have such high drop out rates in later year mathematics courses.

For a system a linear equations it doesn't make sense, because you're multiplying a row by a column.

 

[Mark44]

Yes yes but no one ever actually understands what anything means, which is why you have such high drop out rates in later year mathematics courses.

For a system a linear equations it doesn't make sense, because you're multiplying a row by a column.

The multiplication of a row of a matrix times a column of another matrix (assuming the rows are the same size as the columns) does make sense. It's exactly the same as the dot product of two vectors.

$$\begin{bmatrix} x_1 & x_2 & ... & x_n\end{bmatrix} \begin{bmatrix}y_1 \\ y_2 \\ ... \\ y_n \end{bmatrix}$$
$$=x_1y_1 + x_2y_2 + ... + x_ny_n$$

 

[SteamKing]

Yes yes but no one ever actually understands what anything means, which is why you have such high drop out rates in later year mathematics courses.

Not everyone drops out. Somebody must understand some math some of the time. Otherwise we would be counting on our fingers and drooling at random intervals.

For a system a linear equations it doesn't make sense, because you're multiplying a row by a column.

In a nutshell, that's what matrix multiplication is. The particular entry in the product matrix AB is the result of multiplying the row of A by the column of B, for all combinations of the rows and columns in AB.

 

[RufusDawes]

The multiplication of a row of a matrix times a column of another matrix (assuming the rows are the same size as the columns) does make sense. It's exactly the same as the dot product of two vectors.

$$\begin{bmatrix} x_1 & x_2 & ... & x_n\end{bmatrix} \begin{bmatrix}y_1 \\ y_2 \\ ... \\ y_n \end{bmatrix}$$
$$=x_1y_1 + x_2y_2 + ... + x_ny_n$$

I forgot about that assumption. Now it makes sense.

 

[Mark44]

Matrix multiplication involves one dot product for each element in the product matrix. Matrix multiplication doesn't have anything to do with the cross product.

 

[nmf77]

Matrix multiplication involves one dot product for each element in the product matrix. Matrix multiplication doesn't have anything to do with the cross product.

true, unless you form a skew symmetric matrix from a vector, which is a common way to compute cross products