The trace of a n ×n matrix A ∈ℳn ×n is defined to be the sum of its diagonal entries: tr A=a11+a

step 1 We're asked to prove that if two matrices A and B are similar, then the trace of A is equal to t

The trace of a n ×n matrix A ∈ℳn ×n is defined to be the sum...

The trace of a $n \times n$ matrix $A \in \mathcal{M}_{n \times n}$ is defined to be the sum of its diagonal entries: $\operatorname{tr} A=a_{11}+a_{22}+\cdots+a_{n n}$. (a) Compute the trace of (i) $\left(\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right)$, (ii) $\left(\begin{array}{rrr}1 & 3 & 2 \\ -1 & 0 & 1 \\ -4 & 3 & -1\end{array}\right)$ (b) Prove that $\operatorname{tr}(A+B)=\operatorname{tr} A+\operatorname{tr} B$. (c) Prove that $\operatorname{tr}(A B)=\operatorname{tr}(B A)$. (d) Prove that the commutator matrix $C=A B-B A$ has zero trace: $\operatorname{tr} C=0$. (e) Is part (c) valid if $A$ has size $m \times n$ and $B$ has size $n \times m$ ? (f) Prove that $\operatorname{tr}(A B C)=\operatorname{tr}(C A B)=\operatorname{tr}(B C A)$. On the other hand, find an example where $\operatorname{tr}(A B C) \neq \operatorname{tr}(A C B)$.

The trace of a $n \times n$ matrix $A \in \mathcal{M}_{n \times n}$ is defined to be the sum of its diagonal entries: $\operatorname{tr} A=a_{11}+a_{22}+\cdots+a_{n n}$. (a) Compute the trace of (i) $\left(\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}1 & 3 & 2 \\ -1 & 0 & 1 \\ -4 & 3 & -1\end{array}\right)$
(b) Prove that $\operatorname{tr}(A+B)=\operatorname{tr} A+\operatorname{tr} B$. (c) Prove that $\operatorname{tr}(A B)=\operatorname{tr}(B A)$. (d) Prove that the commutator matrix $C=A B-B A$ has zero trace: $\operatorname{tr} C=0$. (e) Is part (c) valid if $A$ has size $m \times n$ and $B$ has size $n \times m$ ? (f) Prove that $\operatorname{tr}(A B C)=\operatorname{tr}(C A B)=\operatorname{tr}(B C A)$. On the other hand, find an example where $\operatorname{tr}(A B C) \neq \operatorname{tr}(A C B)$.

Key Concepts

Non-cyclic Permutations

While cyclic rearrangements of matrices in a product preserve the trace, this property does not generally extend to non-cyclic permutations. That is, rearranging matrices in a product in a non-cyclic order can lead to a different trace, highlighting the importance of the order of multiplication in non-commutative settings.

Matrix Multiplication and Dimension Considerations

When matrices are not square, special care must be taken in matrix multiplication since the product AB is defined only when the number of columns in A matches the number of rows in B. Even in this more general setting, a version of the cyclic trace property holds, such as trace(AB) = trace(BA) when the products are both defined, illustrating the subtle interplay between matrix dimensions and trace properties.

Commutators and Zero Trace

The commutator of two matrices, defined as AB - BA, always has a zero trace since the cyclic property of the trace ensures that trace(AB) equals trace(BA). This zero trace property of the commutator is significant in areas such as Lie algebras and quantum mechanics, where commutation relations are central to understanding symmetry and conservation laws.

Cyclic Property of the Trace

A key property of the trace is its cyclic invariance, which states that the trace of the product of matrices is invariant under cyclic permutations; that is, trace(AB) equals trace(BA) and, more generally, trace(ABC) equals trace(BCA) equals trace(CAB). This property is instrumental when simplifying expressions involving matrix products and provides insights into the underlying structure of linear transformations.

Linearity of the Trace

The trace function is linear, meaning that the trace of a sum of matrices is equal to the sum of the traces of the individual matrices and that scalar multiplication factors out. This property, expressed as trace(A + B) = trace A + trace B, is fundamental when working with linear combinations and has implications in various applications including quantum mechanics and statistics.

Trace of a Matrix

The trace of a square matrix is defined as the sum of its diagonal elements. It is an important concept in linear algebra because it provides a simple, invariant quantity under certain similarity transformations, and can be used to derive properties of the matrix such as its eigenvalue sums.

Please give Ace some feedback