Why are the columns of Q linearly independent?

[victoranderson]

Why column 1 is M^2*v? How can we know?

Please see attached. Many thanks.

 

[tiny-tim]

hi victoranderson!

if Q^-1MQ = that matrix,

then MQ = Q(that matrix), which is … ? ​

 

[victoranderson]

hi victoranderson!

if Q^-1MQ = that matrix,

then MQ = Q(that matrix), which is … ? ​

I know what you mean
Let that matrix be D
I can find MQ=QD by trail and error
Is there any other method I can use to find the columns of Q without trial and error?

 

[tiny-tim]

hi victoranderson!

(just got up :zzz:)

I can find MQ=QD by trail and error …

it isn't trial and error

what are the columns of QD ?

 

[victoranderson]

hi victoranderson!

(just got up :zzz:)it isn't trial and error

what are the columns of QD ?

it's been a long time..

The columns of QD is [0,0,0]^T, [1,1,-1]^T and [2,3,1]^T
which are $M^3v , M^2v , Mv$ respectively

I am stupid and I still do not understand..

 

[tiny-tim]

hi victoranderson!

for any matrix P with columns A B C:

the columns of PD are 0 A B

the columns of MP are MA MB MC​

so MP = PD (ie P-1MP = D) if 0 = MA, A = MB, B = MC

(so A = MB = M²C and M³C = 0)

so we choose the columns of P to be M² MC and C

(and then we call it Q instead of P)