Similarity - trace and determinant

In summary, if two matrices are similar, then they have the same determinant, although it isn't necessarily true that if they have the same determinant, that they're similar. This tells us that the matrix similar to A must have the same determinant as A. Additionally, if A and B are similar, then tr(A) = tr(B). However, the reverse deduction cannot be made. The cyclic trace property states that tr(ABC) = tr(BCA) = tr(CAB), and tr(AB) = tr(BA). Two matrices are similar if they have the same invariants, such as eigenvalues or coefficients of their characteristic polynomial. However, in this case, none of the given matrices have the same characteristic polynomial coefficients,
  • #1
circa415
20
0
What exactly is the relationship between the trace/determinant of two matrices with regards to similarity. I always thought that if the trace was the same, then there is a possibility that the matrices are similar and if the determinant was the same, then the matrices are similar. On a recent exam we were given three matrices

A
1 0 1
2 3 5
0 2 -6

B
-4 3 4
0 1 2
0 0 1

C
0 0 2
0 4 1
3 5 -2

One of these matrices is similar to A.

I found det(a) = det (c) but trace (a) is not equal to trace (c). Det(B) is not equal to det (a) but trace (a) = Trace (b). Do I have my facts wrong?
 
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  • #2
If two matrices are similar, then they have the same determinant, although it isn't necessarily true that if they have the same determinant, that they're similar. Anyways, this tells you that the matrix similar to A must have the same determinant as A. This is because if A is similar to B, then A = UBU-1 for some U, and:

det(A) = det(UBU-1) = det(U)det(B)det(U-1) = det(U)det(B)det(U)-1 = det(B)

I can't remember any special properties about traces and similarities, but the above should be enough to answer your question anyways.
 
  • #3
If A and B are similar then tr(A)=tr(B) and det(A)=det(B) but absolutely no reverse deduction can be made.

[tex]\left( \begin{array}{cc} 1&1\\ 0&1 \end{array}\right)[/tex] and


[tex]\left( \begin{array}{cc} 1&0\\ 0&1 \end{array}\right)[/tex]

have the same determinant and the same trace (and indeed the same characteristic poly) but are not similar, indeed trace and det are data contained in the char poly and that is the same for all similar matrices. Simillarity is hard to classify simply but can be done with jordan blocks.


For AKG, matrices satisfy the cyclic trace property, and is perhaps weaker than being an invariant of the char poly in that it needs no manipulation other than by simply looking at the entries of the matrices: tr(ABC)=tr(BCA)=tr(CAB), and that tr(ABC)=/=tr(BAC). In fact it is even simpler than that now i come to think of it: tr(AB)=tr(BA)
 
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  • #4
Could you expand on that, matt? What is the cyclic trace property? I'm not sure what is meant by the stuff after that too:

"and is perhaps weaker than being an invariant of the char poly in that it needs no manipulation other than by simply looking at the entries of the matrices"

EDIT: Oh wait, I think I see. The cyclic trace property is that the trace is invariant even if you cycle the matrices A, B, and C around, so as you said:

Tr(ABC) = Tr(BCA) = TR(CAB)

I still don't get the stuff in quotes above though.
 
  • #5
A scalar invarient of a matrix (with respect to similarity) is a function
f:nxn matricies->scalar
such that
f(AS)=f(SA)
for all nxn matrices A and all invertible nxn matrices S
So tr is an invariant
two matrices A & B are similar if
f(A)=f(B) for all invariants
It can be shown that a nxn matrix has n independent invariants thus one could chose their favorite set of n invariants and test those. Independent means that the invarians do not satisfy any relations.
common such sets include
-eigenvalues of A
-coefficients of the characteristic polinomial of A (which det(A) and tr(A) are)
-tr(A^k) k=1,...,nn

So to summerize (~ will mean similar)
A~B->tr(A)=tr(B)
but tr(A)=tr(B) does not imply similarity
n-1 more independent invariants need also be equal.
 
  • #6
Well, we can shor Tr(AB)=tr(BA) by just working out what they are. usign summation convention they are

A_{ij}B_{ji} and B_{ij}A_{ji} which are obviously equal.

hence tr(AB)=tr(BA) is true for elementary reasons, ie we nned not talk abuot characteristic polys or sets of eigenvectors with multiplicity. which was how i first thought of provign the statement until i remembered you don't need to be a smart arse all the time. that's all i meant when i talked about elementary methods; i was talking to myself.
 
  • #7
These are the characteristic polynomials of these matricies.
A^3+2A^2-31A^1+24A^0=0
B^3+2B^2-7B^1+4B^0=0
C^3-2C^2-19C^1+24C^0=0
Since none of these have the same coefficients no pair of these is similar.
 
  • #8
circa415 said:
A
1 0 1
2 3 5
0 2 -6

B
-4 3 4
0 1 2
0 0 1

C
0 0 2
0 4 1
3 5 -2

One of these matrices is similar to A.
In view of the previous posts, the answer seems clear. A is similar to A.
 

FAQ: Similarity - trace and determinant

What is the difference between trace and determinant?

The trace and determinant are two different mathematical operations that can be performed on a square matrix. The trace is the sum of the elements on the main diagonal of a matrix, while the determinant is a single number that represents the "size" or "volume" of the matrix.

How are the trace and determinant related?

The trace and determinant are related in that they both provide information about the properties of a matrix. The trace can tell us about the matrix's diagonal elements, while the determinant can tell us about its overall size and whether it is invertible or singular.

What is the significance of the trace in a matrix?

The trace of a matrix has several important applications in various fields of mathematics and science. For example, in graph theory, the trace of a matrix can be used to calculate the number of closed walks in a graph. In physics, the trace can be used to calculate the energy levels of a quantum system.

How is the determinant used in linear algebra?

In linear algebra, the determinant is used to determine whether a matrix is invertible or singular. If the determinant is equal to zero, then the matrix is singular and does not have an inverse. The determinant is also used to solve systems of linear equations and to calculate the area or volume of geometric shapes.

Can the trace and determinant be calculated for non-square matrices?

No, the trace and determinant are only defined for square matrices. Square matrices have an equal number of rows and columns, which is necessary for these operations to be performed. Non-square matrices do not have a main diagonal or a single determinant value, so these operations cannot be applied to them.

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