Linear Algebra - Orthogonal Diagonalization

[@dngrtegaijin4680]

How did u get sqrt(45) for the last column when doing orthonormal???

[@denny8360]

Clean and easy to understand. Thanks.

[@ayoubmounadi2142]

How do you get that span

[@emathematics3412]

thanksss sirrrrr..its really helpful for my assignment

[@ayoubmounadi2142]

How do we do that row reduction?

[@affaqahmed]

How the √45 come in third eigen vector

[@ramyokash306]

How do we say the columns are orthogonal or not?

[@arnoldyim4765]

You can check if the columns are orthogonal or not by taking their dot product. If the dot product of two vectors is 0, then the vectors or orthogonal.

[@abnerandreymartinezzamudio3366]

If they come from the same eigenvalue, the columns will most likely not be orthogonal, so assume they aren't.

[@sadece_prens]

well, why did we multiply the first eigenvector by 2? I couldnt get it. is it necessary?

[@affaqahmed]

Why the PAP^1 is not equal to A

[@chriswil8252]

How did you know the order of eigenvalues in the diagonal matrix?

[@a_41_pruthvirajkm85]

You can place eigen values in any order but the corresponding eigenvector must be that column of P itself. eg: if you place an eigen value at a11 then eigen vector that we obtained for that eigen value should be considered as the first column of P

[@thenewdimension9832]

Soon u should be on the top 🤗🤗🤗 Keep going

[@ashutosharora2196]

If we have distinct eigenvalues but matrix is not symmetric then also can we orthogonally diagonalize a matrix?

[@abnerandreymartinezzamudio3366]

Yes, if you have distinct eigenvalues, you will have distinct eigenvectors, and therefore the matrix is already diagonalizable

[@allensteephan9651]

Sir can u pls explain how did u calculate the eigenvectors or span when lambda= -3

[@ramyokash306]

What is the difference between diagonalising a matrix and orthogonally diagonalising a matrix?? Is it the same?

[@arnoldyim4765]

When we normally diagonalize a matrix. We write it in the for PDP^(-1). Where the columns of P are the eigenvectors associated with the eigenvalues of the diagonal matrix D. There is no guarantee that the columns of P are orthogonal. When we orthogonally diagonalize a matrix, we make sure that the columns are orthogonal (using the Gram-Schmidt process).

[@thegoofster4686]

@@arnoldyim4765 so you don't need to make the vectors of P orthonormal then?just make them an orthogonal set?

[@arnoldyim4765]

@@thegoofster4686 That's right, you only need an orthogonal set, but having an orthonormal set works too.

[@SADDAMHUSSAIN-mw3cv]

Very thank you respected sir...

[@thenewdimension9832]

Wow .....I am lucky to have this 🥰🥰🥰🥰🥰🥰

[@abnerandreymartinezzamudio3366]

You should call this video: half a college CS semester in 10 minutes.

[@gichuki30]

Here in 2024 I think we should have 3 eigen values

[@jadjonaidi2290]

This video is soooo helpful. Thank you

[@disasterhunter3813]

good

[@ItsPouring]

Thank you! Thank You! Thaaaaank YOU! 💛🤓

[@unknown81000]

Hehe, Godzilla Singular Point

[@mattykrosschell9046]

lol