Complex, Hermitian, and Unitary Matrices

Remember when we talked about complex and imaginary numbers? All that a + bi stuff, it was a while ago. Well that can apply to matrices as well! We've been looking at real matrices thus far, but we can have imaginary or complex matrices if we have imaginary or complex terms as entries. What can we do with these? We can find the complex conjugate, we can find the conjugate transpose, we can check to see if matrices are Hermitian, or unitary, lots of stuff! Let's give it a try.
Script by Howard Whittle
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[@ProfessorDaveExplains]

Whoops! In the checking comprehension for question 1, the top right entry on that matrix should be 2 + i, not 2 + 1. Sorry!

[@chemisthenry3001]

sure, thanks for the Video. May you put more Math videos related to quantum physics.!

[@lamalamassu5337]

Thank you...but I recognize that from the solution.

[@mariyahaxix2360]

It is not square matrix

[@vellabro]

Amazing Playlist ❤

[@b_f_d_d]

​@@vellabroyup

[@allisonphysics6416]

Thanks for saving me once again Prof. Dave! If you ever add to this series can you cover the following: Symmetric bilinear forms, sesquilinear forms, adjoint/ self-adjoint linear maps, the spectral theorem, Jordan Canonical Form, and positive self-adjoint linear maps.

[@charmy3021]

He knows a lot about all kinds of stuff! Professor Dave Explains~~~

[@arberkryeziu8098]

🥴

[@-miracle-]

😂

[@CepheusMappy]

*honk*

[@sanjana3251]

In the comprehension question no.1 in the 1st row 2nd column it was written 2+1 but in the answer you have written it in the form of conjugate transpose 2-i

[@jitenranjan]

I think it's typing mistake it was 1+i

[@jitenranjan]

2+i

[@keshosingh5645]

explained in very simple and lucid style. thank for the video

[@alfandosavant4639]

2nd week of engineering major and I need to recall this. Thank you so much!

[@emlmm88]

Really liking these math videos. You should do a series on modern algebra as well.

[@mohammadamanalimyzada1462]

Thank you so much for the amazing series I have learned many things from it

[@scottrose9260]

Thank you for this, a layman curious about the world, been looking at treatments of this for years, but founds some pearls of wisdom in this video that I didn't know or realize. I suppose they should have occurred to me by now, but your explanations were simple and intuitive and I feel like I gained meaningful insights. Thanks again, looking forward to more of this.

[@Alexandra-he8ol]

I needed linear algebra for better understanding the data science algorithms. Thanks for the well-structured and clear course!

[@harshvisaini2843]

Amazing lecturer. THanks Professor !!

[@sapaephyu6532]

Thx for saving my life, prof.

[@arinalikes5911]

U gotta way to make things lucid, thank you sm ❤

[@michaelzumpano7318]

That was the best explanation I've heard. Thanks. I really appreciated your video where you discussed who you are.

[@mddinar]

This lecture is bloody good.. Thank Prof.

[@harishbalaji658]

Professor Dave it was really pakka, crystal clear

[@theanalyser8456]

This is one of the channels that you just see one video and you instantly subscribe 😊

[@josephckennedy]

This was really well explained! Thanks! I (along with countless others) appreciate this!

[@toxicrow9473]

Could you solve the problems? I am getting 1/3 0 0 1/3 As the result of unitary Matrix It's not identity Matrix tho

[@josephckennedy]

@@toxicrow9473 I’ll give it a try!

[@huythai6210]

thank you so much it's very clear to understand

[@sofiyaprincy]

Professor Dave, I have a topic in my linear algebra and matrix paper called 'Nature of characteristic roots in case of special matrix' . Could you please post on that topic?

[@Im_Your_Proof7]

Thanks a lot Professor Dave💞💞💞💞💞

[@paraskevidimoraga5960]

Excellent, thank you!!

[@InasJaber-e6v]

شكرا ً جزيلاً Thanks a lot

[@rohimakhatun6581]

Thank you sir ...... it's more helpful for me .... thanks a lot

[@fadia217]

Thats great thank you so much ❤ from Jordan 🇯🇴

[@slainiae]

Fantastically explained.

[@jinyoungpung2178]

Crystal clear. Love it!

[@newaz1268]

Thanks, man! Your video is great.

[@송영찬-z6t]

한글자막 누가 달았는지는 모르겠지만 감사합니다 Thank you very much for the Korean subtitles!!

[@nthumara6288]

thank youuu sir

[@anson8570]

Hi. I don't know more things about matrix multiplication. But I have a doubt . Why we are taking conjugate transpose instead of transpose for complex matrix multiplication ? What will happen if we don't take conjugate .

[@can_dospreadgospel5770]

Thanks very very much Professor. You have really thought me a lot. I’m much grateful. God bless you as well 🙏

[@yashaggarwal2822]

Doubt :: i need to find adjoint of Matrix , matrix comprises of Complex elements so What i have to do ? I mean I just have to transpose the matrix or to take its conjugate transpose?

[@areejmohammad7567]

This video is amazing! 👍🏻

[@uditsaxena3844]

Thanks sir very clear

[@ys1940]

Could you make a video about cramer's rule

[@ProfessorDaveExplains]

i did that already

[@ys1940]

@@ProfessorDaveExplains oh really? I didn't notice that😂

[@ProfessorDaveExplains]

check out my mathematics playlist, it's got everything you need

[@bernab]

Yes. it is 2+ i. But anyway. Great video. Thank you Professor Dave. Will there be more videos of Linear Algebra??

[@tpeianlee]

This guy does better than my prof

[@lamees1134]

You are amazing. PLZZZZZZZ keep up the good work up.

[@manjusulekha4042]

Second...🙋very usefull video...thank you sir...

[@moreblessingmushohwe6506]

Thank you Professor Dave!!!! Ps: Found out you came to Cornell a week ago and I didn't get to see you 😥💔💔

[@ProfessorDaveExplains]

I did not

[@moreblessingmushohwe6506]

@@ProfessorDaveExplains my bad 😂. Okay,,, I misheard then

[@danielerimantaadapilah1152]

The first entry in column two is 2+i but not 2+1

[@Tomas-zv2js]

Hey how is the conjugate of that original matrix in the first example not -7 but remains as seven but all the other terms have their signs flipped?

[@ambitionstudypointsurajsah921]

Excellent 👌🏻

[@dayanulalam1678]

4:56 the linear algebra Jesus confirmed

[@momomuscu1261]

Hello , what is the identity matrix in C ?

[@ZaChousenWan]

"Change the sign of each imaginary number on the matrix and turn the rows into columns." vs " Show the Transpose of the Complex Conjugate of the given Matrix." Good teachers vs sadistic teachers.

[@amk1108]

8:15 type on the original matrix. should be 2+i in upper right corner

[@ProfessorDaveExplains]

whoopsie! thanks for catching that, i'll pin a comment.

[@amk1108]

thank you for your videos dave! Keep it Up!!

[@tabindawani6869]

Thanks a lot for your efforts. This video helped me a lot. Lots of love,hope and trust from Indian occupied Kashmir.😍

[@samiransari6707]

What is the value of (i) in unitary

[@dicksonkachamac787]

nice one

[@Yuras20]

If i = sqrt(-1) then -1 = i * i = sqrt(-1) * sqrt(-1) = sqrt( (-1) * (-1) ) = sqrt(1) = 1. That's why I would rather assume i ^ 2 = -1, rather than i = sqrt(-1) because it doesn't lead to such contradictions.

[@ProfessorDaveExplains]

It's not a contradiction. You just said that i^2 = -1, and -i^2 = 1. That's how negative numbers work.

[@damien20ntakirutimana96]

What does orthonormal basis mean?

[@lukasnegumbo1879]

Let’s assume it’s right the way it was given and agree to it being a Hermitian Matrix🙂

[@lalegulbuz3496]

allah anlatsa ancak bu kadar güzel anlatırdı.

[@unlkc]

:D

[@chasethescientistsaturre5009]

Time to go to advance mathematics woohoooooooooooooooo

[@CSBPRAKASHDEEPAK]

Can anyone check q3 in exercise?

[@barfy.man3624]

Inverse short cut doesn't work for complex matrix

[@omarabduljaleelaltuwaijari4754]

thank you...university of Baghdad

[@Recordingization]

Awasome

[@amogh5427]

Easy

[@Wiyt]

What's up with your video view to subscriber ratio?

[@ProfessorDaveExplains]

The only people that are dying to watch a linear algebra tutorial are people who are currently studying linear algebra.

[@Wiyt]

Professor Dave Explains I get that... I just don't understand how other people won't watch. It's such a tiny percentage compared to the people that receive your videos in their subscription box.

[@ProfessorDaveExplains]

Why would they watch if they aren't studying linear algebra? My content is not at all entertaining.

[@Wiyt]

@@ProfessorDaveExplains I'm not sure. Why would a channel with almost half a million subs only have 132 patrons on their Patreon? I know of people with 100k subs that have over 1000+ patreons. Your channel subscribers compared to the amount of people that watch just doesn't add up. I'd assume people would still click on the videos, unless it's just all botted subs. But what do I know?

[@promorizzics]

@Abhay Kamath bad idea

[@Amanda-if4qp]

i th

[@interresistance7614]

2nd one is Not Hermitian 3rd one will be Not Unitary!!

[@bullee]

Wikipedia: In mathematics, a Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. Is this really what you're saying around 5 minutes?

[@bullee]

Ah "hermitian transpose" not the same as "hermitian matrix"