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5h
comment Simultaneous orthogonal diagonalization of two matrices
well, my first comment, about the same eigendirections, made sense when I said it, but a proof is not arriving. Maybe tomorrow.
6h
comment Simultaneous orthogonal diagonalization of two matrices
see these math.stackexchange.com/questions/88022/… math.stackexchange.com/questions/88026/… math.stackexchange.com/questions/1406359/…
6h
comment Simultaneous orthogonal diagonalization of two matrices
in the first edition of Horn and Johnson, Matrix Analysis, it is in table 4.5.15T on page 229. The second edition (2013) does it some other way, I prefer the first. Theorem 4.5.15 is stated on page 228, and case II(b) is expanded on page 231 mostly.
7h
comment Simultaneous orthogonal diagonalization of two matrices
Oh, neither one need be positive definite for there to be a matrix $P$ such as you found, there is an algorithm as long as one of them is invertible.
7h
comment Simultaneous orthogonal diagonalization of two matrices
they would need to have the same eigenvectors. Not the same eigenvalues, or even the same ratio of eigenvalues. Only the directions of the eigenvectors matters. For two by two one can find this out with some square roots at worst.
1d
answered Why is $p_n \sim n\ln(n)$?
1d
comment Why is $p_n \sim n\ln(n)$?
projecteuclid.org/euclid.ijm/1255631807
1d
comment Why is $p_n \sim n\ln(n)$?
there are precise versions in Rosser and Schoenfeld (1962)
2d
comment Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.
@Rowan, the main extra step, not needed for your question, is when all that is left (no explicit squares) is of type $xy$ or, for ease, $4 xy.$ Here we use the trick $(x+y)^2 - (x-y)^2 = 4xy $
2d
revised Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.
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2d
revised Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.
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2d
answered Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.
2d
comment Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.
@CameronBuie, most likely wants a diagonal form. The easy way is Hermite's, but recently a cookie-cutter method has been appearing on MSE, see the links in my previous comment.
2d
comment Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.
Take a look at these: math.stackexchange.com/questions/1388421/… math.stackexchange.com/questions/329304/… (user1551) math.stackexchange.com/questions/395634/… math.stackexchange.com/questions/1388281/…
2d
comment Solving three quadratic simultaneous equations with three variables
the two spheres have the same radius, so the circle of intersection lies in the plane exactly halfway between the centers. Since those centers are the same distance apart, the picture comes out pretty.
2d
revised Solving three quadratic simultaneous equations with three variables
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2d
comment If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$.
related to this en.wikipedia.org/wiki/Vieta_jumping#Example
Aug
26
comment Why should quaternions exist?
In the wikipedia selection I linked earlier, the 4 by 4 matrix that $b$ is multiplying would be $i,$ the matrix multiplied by $c$ would be $j,$ finally the matrix multiplied by $d$ would be $k.$ Note that, as desired, it turns out that $ij=k,$ also $i^2 = j^2 = k^2$ are all minus the identity matrix.
Aug
26
comment Why should quaternions exist?
see the part about 4 by 4 matrices with real entries en.wikipedia.org/wiki/Quaternion#Matrix_representations
Aug
26
comment Bilinear Form Diagonalisation
Thanks. I requested a book from the public library that had some as exercises, by T. W. Korner. I think he always has a positive definite one that is to be sent to the identity, so he will generally be expecting square roots in the matrix $P$ with $P^T A P = I, P^T B P = D.$ Interesting; but it appears we are over the short time when a few people were asking about these topics.