People like you give me hope in myself. I've been struggling in math for the first time in my life (since entering into linear algebra and differential equations) and I was starting to think I was the issue. Nope. Given a good teacher, I.e. you, this stuff is a piece of cake. You're not just teaching people, but you're also restoring their sense of self efficacy, which may have been damaged from poor lecturing. Good lecturers digest or deliver digested content for the student, rather than regurgitating the uncooked, raw formulas and abstractions present in textbooks.
Wow, this is the most wholesome comment I've received this year! I'm super glad you took the time to write that down. Best of luck with your science/ math endeavours, and if you have a question, don't ever hesitate to ask :)
yeah my teacher just gives definitions where maybe he does a 2x2 then gives us homework where it's all 3x3's and have zero fucking clue what everything is or how to do any of it.
This is actually used a lot in Quantum Mechanics! :) ⏱ Timestamps: 0:00 Intro 0:44 Brief background 2:37 How to calculate in general 5:00 Applied to this example 9:35 Trick with diagonal matrices 16:06 Outro
Hello: Thank you for the helpful video. It helps me a lot in my work. And I need your help, on how to find a matrix D (to be found) such that: e^(D ) = C (given square matrix).
Hi there! Thank you for the kind words :) I'm not immediately sure how to find D. But, I would try this Taylor expansion method for a generic 2x2 matrix D = (a,b;c,d), and try to work your way backwards. I might give it a go myself if I find the time. If I do, I will get back to you :) Happy holidays!
What do you mean with fundamental matrix? The matrix exponential is different because it might not be evident to know what it means to have a matrix in the exponent.
That is a technique which is very important in Quantum Mechanics. It's called matrix diagonalisation, and it requires knowledge on eigenvectors and eigenvalues.
In simple terms the exact determination of e^A (without series approximation) for any square matrix A goes as follows: 1.) Subtract your matrix from a same size matrix that (other than zeros) has x's all along the diagonal (top left to down right). 2.) Take the determinant of this new x-dependant matrix. If A is a 2x2 matrix α1 α2 β1 β2 , as you might already know, Det(A)=α1*β2-β1*α2. (The resulting f(x) is called your characteristic polynomial). 3.) Find the roots of said polynomial, so whichever x's make f(x)=0. For a A 2x2 this can be done with the quadratic formula if need be. The roots are called Eigenvalues. 4.) Choose one of the Eigenvalues to put along the diagonal of a new (other than that) zero matrix, like the x's in 1.). Subtract A from it, call it B. Find a (non all-zero) vector, that, multiplied with B, makes Zero. This means solving an equation system, for our A 2x2 example a 2 equation system. This vector is one of possibly many correct Eigenvector in relation to the chosen Eigenvalue. 5.) Repeat 4.) for all the other Eigenvalues, we need to find one for each. In our example A 2x2 there thus will be one other Eigenvector for 2 total to be found. 6.) In a chosen order, slap the eigenvectors next to each other left to right to form a square Matrix V. It will be of equal size as A. Note it. 7.) Invert V. So determine V^(-1)=(1/Det(V))*adj(V) Where for our 2x2 case adj(V), with elements ordered as in 2.), is defined as follows: β2 -α2 -β1 α1 8.) Create a Matrix A_D (the diagonalised Matrix you asked for) by placing the Eigenvalues along the diagonal of an other than that all-zero matrix, keeping the same order of Eigenvalues top left to bottom right as you chose when combining the corresponding eigenvectors left to right in 6.). 9.) As explained in the video, you can now intuitively take e^(A_D), applying the e function to all diagonal entries of A_D (not the zeros!). 10.) Careful: e^(A_D) does not generally equal (e^A). To calculate e^A do the following multiplication: e^A=V*e^(A_D)*V^(-1) Ta-da!
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People like you give me hope in myself. I've been struggling in math for the first time in my life (since entering into linear algebra and differential equations) and I was starting to think I was the issue.
Nope. Given a good teacher, I.e. you, this stuff is a piece of cake. You're not just teaching people, but you're also restoring their sense of self efficacy, which may have been damaged from poor lecturing.
Good lecturers digest or deliver digested content for the student, rather than regurgitating the uncooked, raw formulas and abstractions present in textbooks.
Wow, this is the most wholesome comment I've received this year! I'm super glad you took the time to write that down. Best of luck with your science/ math endeavours, and if you have a question, don't ever hesitate to ask :)
I feel the same !! he's awesome.
@@bhouyem2315 Thank you, it means a lot to me :)
yeah my teacher just gives definitions where maybe he does a 2x2 then gives us homework where it's all 3x3's and have zero fucking clue what everything is or how to do any of it.
Damnnn this 16 minutes video really make me understand everything than 2 hours at my uni classes 😂 nice video thanks bud!!!
You're very welcome! Feel free to share with other students that might benefit from this time saver :D
I love matrishes! 😊
Matrices are such an elegant representation of physical observables in quantum mechanics :)
This is actually used a lot in Quantum Mechanics! :)
⏱ Timestamps:
0:00 Intro
0:44 Brief background
2:37 How to calculate in general
5:00 Applied to this example
9:35 Trick with diagonal matrices
16:06 Outro
i love this explanation!
Thank you! I'm glad you liked it :D Also, I appreciate the comment :)
Muy buena explicación. Felicitaciones
Thank you :D
Hello: Thank you for the helpful video. It helps me a lot in my work. And I need your help, on how to find a matrix D (to be found) such that: e^(D ) = C (given square matrix).
Hi there! Thank you for the kind words :)
I'm not immediately sure how to find D. But, I would try this Taylor expansion method for a generic 2x2 matrix D = (a,b;c,d), and try to work your way backwards.
I might give it a go myself if I find the time. If I do, I will get back to you :)
Happy holidays!
great video thank you!
You are very welcome! :) Thanks for commenting :D
I missed it. So what does the answer exp{[[-1,8],[7,2]]} end up being?
Thanks❤
You're very welcome :))
What’s the difference between fundamental matrix and matrix exponential ?
What do you mean with fundamental matrix? The matrix exponential is different because it might not be evident to know what it means to have a matrix in the exponent.
What does this have to do with lie algebra?
Matrix exponents are used as generators, specifically in Quantum Mechanics.
Is this a part of Linear Algebra?
I'm not sure, it's basically a neat application of Taylor Expansions, used often in Quantum Mechanics. I suppose it is then, yeah :)
@@PenandPaperScience ok!
very good 👍
Awesome! Thanks for taking the time to comment :)
Also, half your 1s look like As ! And hey, that matrix looks like it contains the first 4 digits of e (2.718)!
Yeah, I've had comments about my "1", it's hard for me not to write them like that :}
Well spotted about the first decimals of e :D
Cool! :)
you forgot kayley hamilton
What do you mean? :)
how to turn non-diogonal matrix into a diogonal ?
That is a technique which is very important in Quantum Mechanics. It's called matrix diagonalisation, and it requires knowledge on eigenvectors and eigenvalues.
In simple terms the exact determination of e^A (without series approximation) for any square matrix A goes as follows:
1.) Subtract your matrix from a same size matrix that (other than zeros) has x's all along the diagonal (top left to down right).
2.) Take the determinant of this new x-dependant matrix. If A is a 2x2 matrix
α1 α2
β1 β2
, as you might already know, Det(A)=α1*β2-β1*α2. (The resulting f(x) is called your characteristic polynomial).
3.) Find the roots of said polynomial, so whichever x's make f(x)=0. For a A 2x2 this can be done with the quadratic formula if need be. The roots are called Eigenvalues.
4.) Choose one of the Eigenvalues to put along the diagonal of a new (other than that) zero matrix, like the x's in 1.). Subtract A from it, call it B. Find a (non all-zero) vector, that, multiplied with B, makes Zero. This means solving an equation system, for our A 2x2 example a 2 equation system. This vector is one of possibly many correct Eigenvector in relation to the chosen Eigenvalue.
5.) Repeat 4.) for all the other Eigenvalues, we need to find one for each. In our example A 2x2 there thus will be one other Eigenvector for 2 total to be found.
6.) In a chosen order, slap the eigenvectors next to each other left to right to form a square Matrix V. It will be of equal size as A. Note it.
7.) Invert V. So determine V^(-1)=(1/Det(V))*adj(V)
Where for our 2x2 case adj(V), with elements ordered as in 2.), is defined as follows:
β2 -α2
-β1 α1
8.) Create a Matrix A_D (the diagonalised Matrix you asked for) by placing the Eigenvalues along the diagonal of an other than that all-zero matrix, keeping the same order of Eigenvalues top left to bottom right as you chose when combining the corresponding eigenvectors left to right in 6.).
9.) As explained in the video, you can now intuitively take e^(A_D), applying the e function to all diagonal entries of A_D (not the zeros!).
10.) Careful: e^(A_D) does not generally equal (e^A). To calculate e^A do the following multiplication:
e^A=V*e^(A_D)*V^(-1)
Ta-da!