# Homework Help: Trace prove question

1. Jun 21, 2011

### nhrock3

i need to prove that if tr(A^2)=0

then A=0

we have a multiplication of 2 the same simmetrical matrices

why there multiplication is this sum formula

[iTEX]

A*A=\sum_{k=1}^{n}a_{ik}a_{kj}

[/iTEX]

i know that wjen we multiply two matrices then in our result matrix

each aij member is dot product of i row and j column.

dont understand the above formula.

and i dont understand how they got the following formula:

then when we calculate the trace (the sum of the diagonal members)

we get

[TEX]

tr(A^{2})=\sum_{i=1}^{n}A_{ii}^{2}=\sum_{i=1}^{n}(\sum_{i=1}^{n}a_{ik}a_{ki})

[/TEX]

and because the matrix is simmetric then the trace is zero

why?

i need to prove that if tr(A^2)=0

then A=0

can you explain the sigma work in order to prove it?

2. Jun 21, 2011

### micromass

Hi nhrock3!

This is false. Consider

$$A=\left(\begin{array}{cc} 0 & 1\\ 0 & 0\\ \end{array}\right)$$

then tr(A2)=0, but A is not zero.

What does the exercise say precisely?

3. Jun 21, 2011

### nhrock3

if A is a simetric matrix and if tr(A^2)=0 then A=0

4. Jun 21, 2011

### micromass

Ah, you should have said they were symmetric!

If

$$A=\left(\begin{array}{ccccc} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{12} & a {22} & a_{23} & ... & a_{2n}\\ a_{13} & a_{23} & a_{33} & ... & a_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{1n} & a_{2n} & a_{3n} & ... & a_{nn}\\ \end{array}\right)$$

then what will be the diagonal of A2?? In particular, can you show that the diagonal contains only positive values?

5. Jun 21, 2011

### nhrock3

each member of the diagonal on the A^2 matrix its number of row equals the column number

so the second member in the diagonal is the dot product of the second row with the second column
etc..
so i get this expression
$$tr(A^{2})=\sum_{k=1}^{n}\sum_{i=1}^{n}a_{ki}a_{ik}$$
so because its simetric and equlas zero
$$tr(A^{2})=\sum_{k=1}^{n}\sum_{i=1}^{n}(a_{ki})^2=0$$
so we get a sum of squeres and in order for them to be zero
then each one of them has to be zero
thannkkks :)

Last edited: Jun 21, 2011
6. Jun 21, 2011

### micromass

Indeed, so the k'th member in the diagonal is

$$\|(a_{1k},...,a_{nk})\|^2$$

Do you see that?