6: Eigenvalues: Why det(A-λI) = 0 - Learning Linear Algebra

Full Learning Linear Algebra playlist: • Learning Linear Algebra Finding eigenvectors and eigenvalues. Why do we set the determinant of A-λI to zero? The explanation's right here!
Video on determinants: • 5: Determinants, Inver...
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[@kev7355]

Holy cow. I finally understand this after 20 years. Did this in college back then but never quite understood why det=0.

[@domination2720]

best by far on this topic

[@adityaapoorva2370]

Thank you so much, the explanation was really simple I understood it in one go.

[@SnehilRayal]

Thank you for such lucid explanation 🙂

[@sudipdas4596]

Seriouly, you have tremendous teaching skills. Soved many problems, completed GIlbert strang's videos, had many doubts, this video ironed out the doubts. Love from india

[@RoodJood]

Gilbert Strang and Dr Grinfeld are on another level

[@Junyan-ft5vn]

Thank you!!!

[@jaidonlybbert8853]

Great video. Concise, informative, and well edited.

[@HAL-oj4jb]

Very nice explanation! Is there some easy way to understand why the determinant being equal to zero is a sufficient criterion for an eigenvalue? I always wondered why this is enough to find them (besides checking explicitly).

[@MuPrimeMath]

In the video linked in the description, I talked about why the determinant being zero means that the matrix has no inverse. If the matrix M has no inverse, then Mv=0 has multiple solutions, which is what we need for an eigenvector to exist.

[@springvibes5344]

Thank you

[@jarednair2516]

This is great stuff, thank you!

[@franks8249]

Cool video! Clear explainations.

[@SSNewberry]

Gilbert Strang would be proud.

[@mohamedmaher9747]

Thanks alot.. Simple explanation 👌

[@Sagar-fd5ls]

Great Video!

[@MoonLight-sw6pc]

Nice! Keep going .....

[@hcchang4392]

Thank you !

[@goshikaamarnath2868]

Nice explanation

[@justinchow1644]

only you explained why det=0 is necessary. thanks!

[@freudboyxd3430]

How can i get a t shirt like yours? O.O

[@MuPrimeMath]

Tie dye!

[@lilmaxez]

Let A,B ∈ Mn. If rank(A) < n/2 and rank(B) < n/2, show that 
det(A + λB) = 0 for some complex number λ.

[@NormanWasHere452]

It’s not even that complex to understand why, for some reason it’s just not really explained in textbooks and definitely not by my lecturer lol

[@matthewjames7513]

Great video. But I don't think this is a thorough proof because at 3:16 you assume that that other solutions that aren't v= 0 exist to motivate why the inverse can't exist. Once you know other solutions that aren't v=0 exist, then you can use your logic to show that det(A) =0. Btw It's ofcourse true that other non trivial solutions exist, but you need to prove it first for your logic to work.

[@MuPrimeMath]

This video does not prove the existence of eigenvectors; it proves some properties that eigenvectors must satisfy if they exist. We suppose that v is a nonzero eigenvector and use that derive the matrix conditions.

[@matthewjames7513]

​@@MuPrimeMath Hey thanks for the reply! And thanks again for all your videos. I've been watching a lot of them and I really like your teaching style :) I agree with you that "This video does not prove the existence of eigenvectors; it proves some properties that eigenvectors must satisfy if they exist". But I really think you need to prove non zero solutions exist before, or prove that the eigenvector solution satisfies Ax = λx afterwards. Otherwise there is no benefit. Here's a simple 1D example using your logic to illustrate my point: Find all integer solutions to the equation x/a = 1 where 0

[@casinarro]

​@@matthewjames7513thanks for the example, now I get your point.