# 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!

• Alessandro Iraci

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

• Rookie123

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

• Alessandro Iraci

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

• Rookie123

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

• Alessandro Iraci

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

• Rookie123

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

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• Rookie123

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]

• Alessandro Iraci

Sure, I'll fix it later today.

• Alessandro Iraci

Done!

• Rookie123

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

• Rookie123

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)?

• Alessandro Iraci

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

• Alessandro Iraci

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

• Rookie123

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!

• Rookie123

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

• Alessandro Iraci

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!

• Rookie123

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

• Alessandro Iraci