matrix exponential

Reviewing this 3blue1brown video: https://www.youtube.com/watch?v=O85OWBJ2ayo

The matrix exponential is written as E to the power of a matrix. What do we know about matrix exponentials? We know that they are defined in terms of the Taylor series, so that the Taylor series for $e^x$ is something like

So we can just do that with matrices: we can raise matrices to integer powers with matrix multiplication, as in $e^M = \sum_{k=0}^{\infty} \frac{M^k}{k!}$. This turns out to converge for any matrix, and that's the definition of the matrix exponential.

• I remember a fact that $e^A$ for any matrix $A$ is always a positive definite matrix and that seems related to the fact that with scalars $e^x$ is always a positive real number.
• So apparently the matrix exponentiation comes up in solving ODEs. If you have a differential equation where the the derivative vector is a linear transformation of the position vector, then the solution for the position at time $t$ will be up the form $e^{At}$ where $A$ is the is the linear transformation matrix
• Example of a system like this is in orbit around the unit circle where your velocity is just a 90 degree rotation of your position and at all times so if you take E to the power of the 90 degree rotation matrix then you get the matrix that describes the function from your initial position to your position at time T and it turns out that that matrix it just contains a various sign in cosine terms
• So why is this true well by analogy to the scalar case if we take so let's say we have a system where are drevet if is a is a linear function of our skalar position X then know that that system will have that a solution for that testing will be of the form peter the XT sorry I guess of E to the great art art or solution will be of the form jeeta the si TE where she is yeah somehow a constant that's related to the initial position
• (note: Google voice recognition is so much better than Microsoft):
• Okay I'm thinking about Matrix exponential's and Taylor series and differential equations so we Define The Matrix find the Matrix exponential through a Taylor series we know that the Matrix exponential will solve a class of differential equations in which the in which the rate of change is equal to some linear transformation of the position and now okay so what it what it what does it mean again to solve these equations bated means I guess specifically that I take e to the 84 some value of T I have a concrete Matrix and then I take that Matrix exponential when I get a new Matrix and that Matrix is the mapping from an initial position 2 a final position and okay so that works for rotating around the unit circle because for any amount of time the mapping that we get from an initial position do a final position is always some sort of rotation matrix it's the number of rotations that you will have done after that amount of time so it's it's it's counterintuitive because it's actually we do have a closed form solution and it is just a matrix now I wonder if it's obvious why that Matrix has to be positive definite so I guess what positive definite means is that for any initial position final position is going to be is going to have a positive. Product with d initial position now it seems weird that that could be true if I rotate exactly halfway around the circle then my final position my baby zero -1 and -1 position was 0-1 those things have a negative. Product with each other okay so something is weird
• So I was right out of the Matrix exponential is not always positive definite I think it might be true that the exponential of a symmetric Matrix is positive definite and I think there's a relationship too complex numbers so the 90 degree rotation Matrix is kind of like I the imaginary unit great in that well in the complex plane I guess I owe you can think of I as as the vector and which has zero in the real part and one in the imaginary part and the rotation Matrix is the operation that gets us from the vector that has one in the real part 2 one in the imaginary part which is also multiplying by I so I guess in general traits of a complex number is a vector in the complex plane but through multiplication it's also an operator on the complex plane and the operator is that right is the new thing is equal to I guess you can you can figure it out but it's so a rotation Matrix is not symmetric and I think there's an analogy here between symmetric matrices and real numbers imaginary number to an imaginary number can be negative I believe E2 a real number is always positive of course because every term in the Taylor series is positive and a symmetric Matrix I feel like I should have an intuition for the geometry of symmetric matrices
• So maybe the relevant fact is that every symmetric Matrix is diagonal in some basis so in that bases it acts just as a scaling term and that seemed kind of like a real number because real numbers are scaling terms on the complex numbers