Square root of a matrix

Square root of a matrix: definition and calculation using eigenvalues. What does it mean for a matrix to have a square root?

[danielmilyutin]

With spectral calculus you can define any function of matrix. Either formally or by converging power series. 💪

[Chester]

I don't like to be the one, but: 4:02 the matrix should be [4 -4; -4 4].
Great video doctor ;-)

[michaelnobibux]

A very enthusiastic math teacher! Very rare these days!!!

[mildlyacidic]

Now natural log of matrix Peyam ;)

so we can get sneaky introduction into Lie Groups :P

[FatihKarakurt]

[[1, 2], [2, 1]] is also a solution, so are the negative complements. So, there are 4 solutions in total.

[woowooNeedsFaith]

Your enthusiasm is fun to watch. If I had only √ of it, I would be fine.

[madcapprof]

There is a small mistake in the number of "square roots" of a diagonalizable positive semi definite matrix A.
It would be 2^k, where k is the number of non-zero eigenvalues of A.

[genekisayan]

this is the most excited dude in the whole YT

[GoingsOn]

The way I found one of the square roots of A = [[5, 4], [4, 5]] was by reasoning that it is some matrix R such that |R| times itself is |A| = 9. Which means |R| = 3 or -3. Then I guessed that [[1, 2], [2, 1]] worked, checked its determinant (which is -3), and then multiplied the matrix by itself, and it worked. This method of diagonalization you have demonstrated is a lot more intuitive!

[piguyalamode]

You can do this other times, such as a rotation matrix a=[[cosø, -sinø], [sinø, cosø]] -> √a = [[cos(ø/2), -sin(ø/2)], [sin(ø/2), cos(ø/2)]], but it is unclear when you can do this.

[etienneparcollet]

You can also do this for other matrices but generally the answer is non-double. You need tu use sub-spaces stability arguments to get the answers.

[eneapane]

Try cubing it and call it matrix triology😂

[sam-kxty]

I love the way you combine knowledge of both linear algebra and calculus to help solve knew math problems every time, I sure hope I can be as calm as you later in future and thanks for this video .

[sushmitamukherjee]

0:33 it should have been "if and only if y>=0" o0therwise
Square root of 4 can be -2 according to ur definition

[shmuelzehavi]

There is another positive solution, which is the matrix: [1 2 ; 2 1].

[silasrodrigues]

Find a polynomial p(x) such that p(y) = f(y) for all y, where y are eingenvalues of A, so p(A) = f(A), where f(x) is any function defined at the points x=y.

[tdiaz]

That is so cool! Thanks for this linear algebra refresher.

I do have a question, how would you go about finding the other matrices that give B^2 = A?

[danielfisher]

Thank you for the video. At 1:13 you mention there are N^2 square roots. Are there not 2^N square roots, obtained by independently choosing between the positive and negative square roots of each of the N eigenvalues?

[patrykszlufik]

Doesn't it leave like n^2 (for nxn matrix) possibilities of square roots (just by this definition; over complex numbers?). Also why eigenvalues and not determinant afterall?

[theproofessayist]

Wow diagonalization really does wonders - and now you can generalize this for any fractional power of a matrix.