A subtlety in property #1 in identifying a ref or rref matrix
My linear algebra book says for property #1 as follows:
1) If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leading 1.
Then I came up with my own matrix in my mind and started questioning property #1.
2 1 5 1
0 0 1 0
Would the number 2 in the first row of this matrix be the "leading 1" ...after all the textbook did say "first nonzero number"?
No, that is because that matrix is not in reduced row echelon form. You need to divide the first row by 2 and then subtract 2.5 times the second row from the first row to make it row reduced, at which point you get:
1 0.5 0 0.5
0 0 1 0
Which is rref as it is in row echelon form, has leading coefficient 1 in every row, and every leading coefficient is the only nonzero entry in its coumn.
The definition seems to not be universal, though, according to Wikipedia some authors let a row echelon form matrix have leading coefficients that aren't 1, while others require them to be 1. Either way however a reduced row echelon form matrix (like the one above in my post) must have leading coefficients of 1 as that representation is unique.
“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”
No, that is because that matrix is not in reduced row echelon form.
Oh now I get it. You have to reduced it the form before you can apply the 4 properties as a confirmation. That does make more sense. Thanks!
Well rref doesn't come up much in game dev anyway, since the matrices are so small.
Well rref doesn't come up much in game dev anyway, since the matrices are so small.
Thanks for the note. By the way, Is matrices generally used only for 3D games? The only time I used the idea of matrices was only when I needed to use arrays.
Well rref doesn't come up much in game dev anyway, since the matrices are so small.
Thanks for the note. By the way, Is matrices generally used only for 3D games? The only time I used the idea of matrices was only when I needed to use arrays.
4x4 matrices are used all over advanced physics, especially relativity where you need it pretty much for the same reasons as computer graphics (a 3D affine space). Higher dimension matrices are used in various branches of applied statistics, etc... so, no, matrices are not only for CG, they have a wide range of uses across many disciplines.
“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”
Well rref doesn't come up much in game dev anyway, since the matrices are so small.
Thanks for the note. By the way, Is matrices generally used only for 3D games? The only time I used the idea of matrices was only when I needed to use arrays.
4x4 matrices are used all over advanced physics, especially relativity where you need it pretty much for the same reasons as computer graphics (a 3D affine space). Higher dimension matrices are used in various branches of applied statistics, etc... so, no, matrices are not only for CG, they have a wide range of uses across many disciplines.
Interesting. Thanks.
