Linear Algebra: Quadratic Forms and Matrix Norms

Please answer all questions (3a, 3b, 3c, and 4) showing as much working out as possible please. 

The questions are on quadratic forms, orthogonal matrices, max and min values, matrix norms. 

Thanks!

  • "Matrix norm" can have multiple different meanings. Which one do you mean exactly?

  • This was just in reference to the text in question 4, which reads "compute the matrix norms of..."

  • Yes, but matrix norm might mean multiple things. What is your definition?

  • Alessandro, please see my most recent quesiton posted titled "matrix norms_ Not A Question". The document attached to this gives my definition of matrix norm - thanks

  • Noted, thank you! Also I think you can attach files here too, no need to open another question. Maybe check with the admin.

  • I'll look into how to do that! Thanks Alessandro

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  • Alessandro, in question 3a, can you explain how you actually arrived at the entries at the very beginning, where you have A = [the matrix entries]

  • Sure, I'll fix it later today.

  • Done!

  • Hi Alessandro, for question 3b you say to "set (x1, x2, x3) equal to -2, 1, and 3 times itself". You go on to do this for 1 and then for 3, but not for -2, is there a reason for this? Thanks in advance

  • Also, in question 3b, when setting (x1, x2, x3) equal to 1 times itself, and you arrive at "x1 = 2x1 - x3", which is "x1 = x3", should the eigenvector associated with this not then be (1, 0, 1) in stead of (1, 2 , 1)?

  • Whoops, I must have lost it in the post-processing, I'll fix it immediately.

  • Fixed, and clarified the point in your previous question. Sorry about the inconvenience! :)

  • Thanks Alessandro, appreciate that! I have one last question: in question 3b for eigenvalue -2, we have "= -x1 + 5x3, which gives 5x1 = 5x3". Where did you get that last part "5x1 = 5x3", specifically the "5" in "5x1" - thanks!

  • also, for 4, should the answer be the second larger root, instead of the first, smaller root?

  • In the part for -2, the equation reads "4x1 = ... = -x1 + 5x3", so that gives "5x1 = 5x3". For 4, yes, I wrote the right thing above and then copied the wrong thing below. It's fixed now!

  • Thanks for the explanation and the quick response, really appreciate it! I'm accepting your answer now, thanks

  • Cool, sorry about the typos! Glad to have helped!

The answer is accepted.