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Mike, the Mathematician
Добавлен 22 янв 2023
Welcome to Mike the Mathematician's channel! Mike Dabkowski is a professor of Mathematics at the University of Michigan-Dearborn (he went to school there too!). Mike loves mathematics and loves sharing it with others. Mike did some graduate studies at Michigan State and earned a Ph.D. at the University of Wisconsin (On Wisconsin!).
Mike is the Director of Math Corps U(M)-Dearborn (@MathCorps), coach of the UM-Dearborn Math Competition Team, and co-advisor to the Actuarial Math majors at UM-Dearborn.
Mike is the Director of Math Corps U(M)-Dearborn (@MathCorps), coach of the UM-Dearborn Math Competition Team, and co-advisor to the Actuarial Math majors at UM-Dearborn.
Similar Matrices
We define what it means for two matrices to be similar. Namely, A is similar to B if there is an invertible matrix P such that PA = BP. This is slightly different than the notion of equivalent matrices (in this case there are two invertible matrices P and Q such that PA = BQ). We prove that similarity is an equivalence relationship. We also show that if two matrices are similar, then they have the same characteristic polynomial and hence the same eigenvalues. Since multiplying by an invertible matrix does not distort the rank, two similar matrices also have the same rank.
#mikethemathematician, #mikedabkowski, #profdabkowski
#mikethemathematician, #mikedabkowski, #profdabkowski
Просмотров: 33
Видео
Orthogonal Complements of Null Spaces of Matrices and their Transposes
Просмотров 362 часа назад
We consider the relationships between null spaces and images of matrices and the orthogonal complement of their transposes. The structure of the inner product and transpose operators allows us to characterize the orthogonal complement of null spaces and images in terms of the images and null spaces of the transpose respectively. We use the fact that the orthogonal complement and transpose are i...
The QR Matrix Factorization: A 4x3 Example
Просмотров 564 часа назад
We give an example of computing the QR factorization of a 4x3 matrix. The QR factorization takes a matrix and applies Gram-Schmidt to the columns. Since Gram-Schmidt preserves the spans of the columns in the original order of the product of the columns of A with the transpose of the Q (the matrix whose columns are the result of Gram-Schmidt) will result in an upper triangular matrix. Our exampl...
Orthogonal Matrices are Isometries
Просмотров 1977 часов назад
We define an n by n matrix, A, to be an orthogonal matrix if A^T is A^{-1}, that is, the transpose of the matrix is its inverse. When this is the case, the columns of the matrix are an orthonormal basis of R^n. These matrices preserve the length of any input vector and we therefore call them isometries. #mikethemathematician, #mikedabkowski, #profdabkowski
Gram-Schmidt and the Legendre Polynomials of Small Degree
Просмотров 1899 часов назад
We apply the Gram-Schmidt algorithm to the set of polynomials {1, x ,x^2} with respect to the standard inner product on the space of continuous functions on [-1,1]. The orthogonalization procedure results in the first three Legendre polynomials (up to a normalization). These polynomials occur in many partial differential equations as we will see if further videos. #mikethemathematician, #mikeda...
Determinant of the Minimum and Maximum of the Row and Column Number
Просмотров 3712 часов назад
We prove that the determinant of the matrix whose entries are the minimum of the row and column number is one. We also prove that the determinant of the determinant whose entries are the maximum of the row and the column number is n*(-1)^(n-1) by induction. In both arguments we use the row and column reduction properties of matrices to simplify the calculations. #mikethemathematician, #mikedabk...
Computing a Determinant by Considering a Special Product
Просмотров 4412 часов назад
We compute the determinant of a block symmetric matrix by considering a special product. The determinant of a product is the product of the determinants, so if we can find a matrix which replicates a similar form to the original matrix, we can derive a formula for the determinant of the original matrix. #mikethemathematician, #mikedabkowski, #profdabkowski
Computing a Determinant Containing Variables using Differentiation
Просмотров 5914 часов назад
We give an example of computing a determinant which contains variables. We differentiate the determinant using the differentiation product rule for determinants to show that it is constant in one of the variables. This allows us to choose a value of that variable which makes computing the determinant elementary. #mikethemathematician, #mikedabkowski, #profdabkowski
Circulant Matrices and their Determinants
Просмотров 10316 часов назад
We introduce circulant matrices. These are matrices whose entries are invariant when you simultaneously shift the rows and columns by the same amount. These matrices arise when considering translation invariant operators and the Discrete Fourier transform. Circulant matrices are diagonalized by the Fourier basis, so we use this fact to compute the determinant of a circulant matrix in the 4x4 ca...
Orthogonal Complements and Orthogonal Projections onto Subspaces
Просмотров 5319 часов назад
We discuss the orthogonal complement of a subspace of an inner product space. This is a subspace in which every element is orthogonal to the original subspace. With this notion, we are able to construct more general projections onto subspaces. We prove that a finite dimensional space can be decomposed into the direct sum of a subspace and its orthogonal complement. #mikethemathematician, #miked...
Gram-Schmidt Orthogonalization
Просмотров 5421 час назад
We discuss the Gram-Schmidt procedure for taking a given collection of vectors and finding an orthonormal set of vectors with the same span. We achieve this my using the subtracting off the orthogonal projections onto the subspaces of the previously constructed vectors. We give an example of how this procedure works for a set of three vectors. #mikethemathematician, #mikedabkowski, #profdabkowski
Plancherel and Parseval Identities for Orthonormal Bases
Просмотров 59День назад
We prove both the Plancherel and Parseval Identities for Orthonormal Bases in a finite dimensional vector space. The unique representation of a vector as its Fourier coefficients dot the orthonormal basis allow us to conclude that the square of the norm of a vector is the sum of the squares of the Fourier coefficients (Plancherel's Theorem). The representation allows us to conclude that the inn...
Orthonormal Bases and Fourier Representations in Euclidean Space
Просмотров 133День назад
We discuss orthonormal bases of Euclidean Space. We prove that an orthonormal collection of vectors is always linearly independent and therefore a sufficient number of them will form a basis of a finite dimensional vector space. An immediate consequence of the linear independence argument is the the fact that the coefficients in the orthonormal basis are easy to compute; they are the Fourier co...
Orthogonality in Euclidean Space: The Pythagorean Theorem and the Orthogonal Decomposition
Просмотров 51День назад
We prove the Pythagorean Theorem and derive the Orthogonal Decomposition in high dimensional Euclidean Space. The bilinear structure of the inner product together with the definition of orthogonality will help us understand the geometric importance of an inner product on a more general space. The orthogonal decomposition is equally important, as it will allow us to define projection onto a subs...
The Cauchy Schwarz Inequality in R^n
Просмотров 97День назад
We prove the Cauchy-Schwarz Inequality for vectors in n-dimensional space. This is one of the most important inequalities in mathematics and it is used a a common tool on my mathematical competitions. The proof employs the convexity to find the global minimum of a function involving the norm. Balancing the parameters in the problem yield the inequality. #mikethemathematician, #mikedabkowski, #p...
Dot Product and the Parallelogram Law in High Dimension
Просмотров 8814 дней назад
Dot Product and the Parallelogram Law in High Dimension
The Eigenvalues of AB are the Eigenvalues of BA
Просмотров 7114 дней назад
The Eigenvalues of AB are the Eigenvalues of BA
The Identity Matrix Plus a Rank One Perturbation
Просмотров 7614 дней назад
The Identity Matrix Plus a Rank One Perturbation
The Eigenvalues of a Special Tridiagonal Matrix
Просмотров 17614 дней назад
The Eigenvalues of a Special Tridiagonal Matrix
Dimension of Subspaces of Euclidean Space
Просмотров 18014 дней назад
Dimension of Subspaces of Euclidean Space
Bases for Subspaces of Euclidean Space
Просмотров 7421 день назад
Bases for Subspaces of Euclidean Space
Rotating an Ellipse: The Principal Axis Problem
Просмотров 10521 день назад
Rotating an Ellipse: The Principal Axis Problem
Rotating a Hyperbola: The Principal Axes Problem
Просмотров 7321 день назад
Rotating a Hyperbola: The Principal Axes Problem
Gershgorin Circles and Diagonally Dominant Matrices
Просмотров 16321 день назад
Gershgorin Circles and Diagonally Dominant Matrices
Orthogonal Diagonalization: An Example
Просмотров 9021 день назад
Orthogonal Diagonalization: An Example
Invariant Subspaces of Projections, Reflections and Rotations in Two Dimensional Space.
Просмотров 11221 день назад
Invariant Subspaces of Projections, Reflections and Rotations in Two Dimensional Space.
The Construction of Brownian Motion The Sample Paths are Almost Surely Continuous
Просмотров 7921 день назад
The Construction of Brownian Motion The Sample Paths are Almost Surely Continuous
The Construction of Brownian Motion: The Series Converges Almost Surely
Просмотров 8621 день назад
The Construction of Brownian Motion: The Series Converges Almost Surely
Can u make Laplace equation in polar coordinates for three dimension??
can you also do about pcl(principal component analysis ) and the concept of distance please
@yonathanmussie9323 You bet! Those videos will come out soon! I will comment when they are ready!
Where was this when i was in calc 4 😢
@peteedwards1461 Sorry I wasn't there earlier! Thanks for watching! I am here to help now!
Awesome bro! May I ask you what happens if we put M - N?
@marcelosebastian5442 The support of M-N will be the entire collection of integers (both positive and negative), so it typically isn't considered as Poisson random variables are usually used to model nonnegative counts. Thanks for watching!
Thanks for videos Sir. I newly saw your channel and it is amazing. Can you write the name of the books that you are following for that playlist to description of playlist. Which book do you use for linear algebra?
@volkandemir6353 Thanks for watching! I use my own notes, but I reference Linear Algebra Done Right by Axler and Linear Algebra Done Wrong by Treil!
So why haven't mathematicians APPLIED this inequality to Archimedes' n-gon approach? If Isoperimetric Inequality is true, it is IMPOSSIBLE to converge to a perfect circle from a non-. Mathematicians have yet to apply what this inequality tells us about circle vs. non- comparisons.
Hi isn't it easier to compute the matrix by seeing that if you divide it into 2x2 blocks, each of these blocks embed in a 2 dimensional algebra a + b.e where e² =1. This algebra is commutative. So the determinant of the original matrix corresponds to: (a +b.e)² - (c +d.e)² = (a + c + (b+d).e)(a - c + (b-d).e) Now, the determinant now corresponds to the determinant of this product (by reembedding it into the matrix algebra) which is much easier to do, and to factorize.
You can also just compute the final determinant through the norm (this is a finite dimension algebra extension after all, so the norm corresponds to the determinant of its embedding) (The only automorphism which fixes the base field is e -> -e.)
You really need to slow down your speech. Or, more importantly, PAUSE between concepts so listeners have the opportunity to process what you say.
@joekemp83 Thanks for the comment! My students tell me the same thing! It is something that I will work on in the future! Thanks for reminding me!
Apply this to Archimedes' 3rd postulate in his 'CIRCVLI DIMENSIO' P > c > p for circumscribed n-gon perimeter P & insc. n-gon perimeter p & circle circ. c. If isoperimetric inequality is true, Archimedes' 3rd postulate is unsound. In fact, all non-circular approaches are unsound due to the inequality. To solve for π while/as preserving isoperimetric equality, you have to treat each π/4 as n = 1 such that n = 4... not n → ∞. Do this by rolling the unit diameter circle on a flat plane surface y = -1/2 & use algebra to find the linear distance its origin travels per π/4. Hint: if you expand a second circle from the first by an areal factor of π/4 per rotation, you will end with a surrounding annulus of area π. Use the width of this annulus w & its orthogonality to π/4 length to construct a right triangle sides w, π/4 bounded by a hypotenuse of 2r = 1. pdfhost.io/edit?doc=7507a819-627f-4eee-beaa-441e685e5157
why is it so dark in there?
wow, how do u do that?
Excellent video hehe. I am now a follower of yours from now on. Btw, why does your random variables have overbars and underbars?
Excellent video
@darrenpeck156 Thanks so much!
you sound alot like MatPat! thank you for the video.
Thank you! All the videos and articles in my language, that I've seen, use some rocket science as explanation, and this video was very easy to follow along.
Is it also possible without the computation of the eigenvalues ? Maybe just comparing the signs of f_xx and the determinant of Hess_f ?
@alidaqa2738 Yes it is, but this method works in any dimension. The standard formula for the Hessian only works in two dimensions.
does the "Vk - sum(proj Vk)" means that the all the vector components , not in the direction of phi_k , is eliminated/deleted ?
Yes! You got it!
Geeeee, no cut at all? Amazing, man!
If anyone is interested in a note of this lecture, pls check out www-users.cse.umn.edu/~jwcalder/5587F16/lecture4.pdf
How to solve 0x=0 equation thanks
@armanavagyan1876 Every real number is a solution! Thanks so much for everything!
I wanted to say U immense thanks for publishing regularly new VIDEOS)
@armanavagyan1876 Of course! I am trying my best! Thank you so much Arman!
It is a symmetric random walk
@rajkro9615 Indeed! Thanks for watching!
How do you define ”every elements of 0, 1”? What is an ”element”?
@MrFedX Every "element of [0,1]" is any real number between 0 and 1. Thanks for watching!
@@mikethemathematician thank you. I’m wondering, how can one be allowed to list discrete elements if the real numbers are a continuum? For example, when numbers follow sequentially like 0.999… and 1 they are defined as the same real number. That would imply that no two concrete numbers can exist next to each other and it’s just limits all the way down. :)
Magic Mike does it again! 🎉😊
@punditgi Thanks so much Ezra!
I have seen anyone proving this with vectors 😅.. Nice video actually tho!!
@ARN48411 I know cool right? I will post some more videos on ideas like this (Ceva's Theorem in general)!
@@mikethemathematician Sir, Can you please make a video explaining why do vectors actually works on Forces.. I mean.. I can show it works on displacement but I don't why it actually works on all vectors quantity.. Can you please prove it???? 🙏
Awesome, this video helped me a lot 👍
@NurbolBelyal Thanks so much!
Hi, many thanks, you made something click for me! Say, is it standard to teach the entire proof of computing a Jordan form? It's very pretty and interesting to grasp, but given the practical uses, it seems so redundant.
@hhhhheeeelllllllllooooo Thanks so much!
Your channel really came clutch during my endsem exams. Love your work
Magic Mike! 🎉😊
@punditgi Thanks so much Ezra!
Mike, The Mathematician Or Whot?!! 🔥
Thanks! @ShanjayS-zw3ox
Thanks it helped a lot❤
I love singular differential equations! @PhysicsMath-my3cj
bro are you writing backwards
Pfft. Child's play
@ungarlinski7965 There is a fun problem related to this that I wrote a paper on! imagine that the sub and super diagonals have entries which are geometrically decreasing. Do the eigenvalues of the nxn matrix and (n-1)\times(n-1) matrix interlace? Cool stuff!
@@mikethemathematician Know that I was only kidding earlier. So the paper answers that? Is that your own question? I only know of tridiagonal matrices from Terwilliger at Madison.
Thank you!
@WeaponBalalaika You are welcome! This is so cool right?
What I don't understand is the intuition behind it. I understand the mechanics, but not the intuition.
@croncoder862 There are a ton of differential equations videos that I have made which have continuous versions of this!
@@mikethemathematician thanks will checkout
Thanks for the video sir.
@croncoder862 You are welcome! This is a cool equation. If we don't assume continuity, the axiom of choice generates lots of other solutions!
Congratulations to University of Michigan Dearborn Actuarial Science Program that receives Casualty Actuarial Society and Society of Actuaries recognition. I am studying Applied Statistics and Economics
excellent explanation, it is helping very much my studies
W(t+s) - W(t) = N(0,s) is independent of W(l), l <= t, does this imply that the brownian motion is a wide-stationary process?
2:01 cant be again equals 0<=t<=1 the correct is 0<t<=1 same with t>=1
Surely your title should say "dominant", not "dominate".
@godfreypigott Thank so much! I fixed it! I hope that you enjoyed the video!
@@mikethemathematician Haven't watched it yet. Saving it up for Jim Beam o'clock.
Thanks for the vedio.
@dfsfgdg Thanks so much!
Great video! What I don’t understand is how dedekind cuts create the real numbers if they are subsets. If you define a set a a collection of subsets, and those subsets only contain rationals, how to the reals arise? Are the reals a set of all supremums of the dedekind cuts? I was also wondering how we can create a dedekind cut for the cube root of two before having defined what a cube root means in the reals.
@obikenobi3629 Great question. In every construction of the real numbers, the definition is a bit surprising. In the Cauchy sequence definition, real numbers are defined as Cauchy sequence of rationals modulo Cauchy sequences converging to zero. This sequence and all of its representatives can be thought of as a subset of Q as well!
I really appreciate your videos and effort.
@Yoyo47333 Thanks so much!
the correct solution
@PaoloACostantino Thanks Paolo! Thanks for the notes too!
Thak you so much! This is very thorough. I was struggling to understand it from an oversimplified PDF my course gave me.
@carolinadama Thank you so much!
thanks a lot
@birkacsorumvar Thanks for watching!
really helped my understanding, really appreciate it!!
@teoo8816 I am very glad! Thanks!
Hi, I gost lost at the part that you use the integrating factor. Can you explain? Thanks!
Explaining level sets whilst writing backwards.
@dalelinney8437 I know mad skills right? Thanks for watching!
Hi Mike, loved the video. A short note: in the first example where you expand the cos^2 x, shouldn't there be a co-efficient of 1/2 before the summation sign in the term of n=1?
@rounakchakraborty8074 Thanks so much! Good catch! I forgot to carry the 1/2 along for the ride! Please make the appropriate corrections in your notes! Thanks again!