MHB Proving Positive Definite Scalar Product for $n \times n$ Matrices

rputra
Messages
35
Reaction score
0
Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product:
$$\langle X, Y \rangle = tr(X Y^T),$$
and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are
(1) $\langle X, X\rangle \geq 0$ and
(2) $\langle X, X\rangle > 0$ if $X \neq 0.$
I am lost on proving these two properties, any help or hints to prove them would be very much appreciated. Thank you before hand for your time and effort.
 
Physics news on Phys.org
Hi Tarrant,

A good place to start would be with the definition of the trace for an $n\times n$ matrix. Do you recall what this was?
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'
This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
Back
Top