Example of Spectral Theorem (3x3 Symmetric Matrix)

Linear Algebra: We verify the Spectral Theorem for the 3x3 real symmetric matrix A = [ 0 1 1 / 1 0 1 / 1 1 0 ]. That is, we show that the eigenvalues of A are real and that there exists an orthonormal basis of eigenvectors. In other words, we can put A in real diagonal form using an orthogonal matrix P. (Eigenvalues and eigenvectors for this A are found in the video "Eigenvalues and Eigenvectors.")

[danielvazquezguevara]

You explained all this in less than 8 min while my professor took over 2 hours and yet did not get it. Thank you so much!

[mrrasheed]

i love you dr.bob you saved my life tonight <3

[brookerthebaker]

You remind me of my Dad, but you're a way better teacher than him and you won't get frustrated if I have to watch this video multiple times. Thanks!

[MathDoctorBob]

Your welcome! That's one big plus about these. (Based on popular opinion, is your dad Henry Rollins or Nic Cage?)

[MathDoctorBob]

My students talk right over me, no fear. Exam 1 may change that.

[MathDoctorBob]

Check out the Positive Semidefinite Matrix 1-3 videos in the Matrix Theory playlist. I do square roots of matrices, Spectral Theorem for C, and some matrix factorizations. It may be easier to find in the master lists on the website.

If you need more, let me know.
- Bob

[sitamahalakshmibharatula]

really appreciable thanks for an amazing video

[FallenLovesX]

Whenever doctor bob has the stick, I look at his biceps and fear that if i talk in class, he would smack me with it....

[arisoda]

but... applying Gram-Schmidt will change the eigenvector into another vector, which is not necessarily still an eigenvector, right...?

EDIT: ah so as long as Gram-Schmidt is used on one eigenspace at a time, the resulting vector is still an eigenvector, even to the same eigenvalue of the used eigenspace

[gegiojonjongegio]

I tried with another matrix, but the P matrix I got from the orthonormal basis did not satisfy transpose(P)=inverse(P)
(the basis is not wrong, the products of two different vectors were 0 and the porduct of one vector with itself was 1)

[pedrooliveirasantos]

dunno if you'll see this comment. but i dont understand how after you aply gram-schimidt to the eigen vector, the resulting vector is still an eigen vector

[ricardocastillo]

Shouldn't the unit vector or orthonormal basis be sqrt(9) instead of sqrt (6) for the third column in P? Also the middle in the ONB shouldn't be -1/sqrt(9)?

[b-systo]

Salut ms merci bcp pour cette l'explication mais j'ai un qst a qoui t égal SP(T, S)=?? Tq T et S ds BL(x)
SP :spectrale

[monyi]

I want to know;how to calculate the eigenvectors.