Example of Diagonalizing a Symmetric Matrix (Spectral Theorem)

Linear Algebra: For the real symmetric matrix [3 2 / 2 3], 1) verify that all eigenvalues are real, 2) show that eigenvectors for distinct eigenvalues are orthogonal with respect to the standard inner product, and 3) find an orthogonal matrix P such that P^{-1}AP = D is diagonal. The Spectral Theorem states that every symmetric matrix can be put into real diagonal form using an orthogonal change of basis matrix (or there is an orthonormal basis of eigenvectors).

[polishhammer]

You use basic methods which makes life easier! When I graduate in a year and become a math teacher myself, I will teach very similar to this. Listing the goals, list of steps you need to take etc. Shocked more teachers don't teach this way. It's so effective for learning!

[themindfulmint]

@MathDoctorBob And sorry! New to this subcribing thingy. Couldnt figure out the playlist.

Could you do some topics on Infinite Series? " D'Alembert's Ratio Test" "Logarithmic Tests" "Rabbie's Test" please?
P.S. You are a great help sir.. I am an engg student and your videos are exactly what I need and at the right time!
Thanks for helping out us. Maths has always been a scary subject!! But you made it very easy!
And like your voice :)
Keep posting!! * Thumbs up *

[SeraphisQ]

This is the perfect video of how diagonalization of symmetric matrices work!

[MathDoctorBob]

@pz0utable You're welcome, and thanks for the kind words! - Bob

[MathDoctorBob]

Thanks! I annotated. Usually the previous section will be on orthogonal matrices and orthonormal bases. Of course, it takes time to put together first time through, especially as things get more abstract.

Diagonalization doesn't depend on symmetric - we can diagonalize [0 -6 / 1 5]. With symmetric, the eigenvectors for different eigenvalues are orthogonal, which is extremely useful. - Bob

[SMABackup]

Thanks Bob! This really cleared some stuff up!

[MathDoctorBob]

@estro2222 You're welcome, and thanks for the comment. Yes - the corresponding result for complex matrices requires Hermitian matrices (replace transpose with conjugate transpose). We would want to develop the theory of Hermitian inner products first.

[MathDoctorBob]

It's worth it. There are some huge ideas here, and Spectral Theory leads to big math and physics results. Where Fourier series/transforms come from begins here. - Bob

[pzutable]

Amazing video and explanation! Thanks for the help.

[MathDoctorBob]

@26877271 There are about five videos on Cayley-Hamilton in the Matrix Theory playlist. There are also about five videos on Eigenvectors and Diagonal Matrices in the Linear Algebra playlist.

Is the second topic Hermitian matrices? - Bob

[estro]

Thanks, great video.
I think it worth to note that we talk here about real symmetric matrices, because when dealing with complex symmetric matrices [latex]A^t = A^{-1} [/latex] is not enough, am I right?

[coldfusion]

Good explanation. But i think you should maybe put some more explanation where the squrt 2 comes from. The book kinda skips that part as well lol. So i'm curious; let's say the matrix is not symmetric; would it still be possible to diagonailze the matrix?

[jackyfish]

Thx Dr.Bob for your easy understanding math. video, its really helpful. A College student Form Hong Kong.

[Martina]

I didn't understood if it's my original matrix or my vectors that have to be symmetrical, which one is it?

[themindfulmint]

Could you please explain the topics :
Caley Hamilton Theorem
Hition Matrix
Eigen Vectors

[froxtroxproxoxox]

question: Hey Bob, I saw the video talked about inner product a lot but I am a bit lost in where is the inner product involved?

[ChandlerMakesVidya]

tfw your gym coach is also your linear algebra professor

[MathDoctorBob]

@sameemas2 You're welcome! Good luck with exams. - Bob

[MathDoctorBob]

You're welcome! Good luck on your exam.

[MathDoctorBob]

We want an orthonormal basis, so we need orthogonal and lengths normalized to 1.
sqrt(1 + 1) = sqrt(2). - Bob