12.5: Special operations on matrices
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In this section, we define three important operations on matrices called the transpose, conjugate transpose, and the trace. These will then be seen to interact with matrix multiplication and invertibility in order to form special classes of matrices that are extremely important to applications of Linear Algebra.
A.5.1 Transpose and conjugate transpose
Given positive integers m,n∈Z+ and any matrix A=(aij)∈Fm×n, we define the transpose AT=((aT)ij)∈Fn×m and the conjugate transpose A∗=((a∗)ij)∈Fn×m by
(aT)ij=aji and (a∗)ij=¯aji,
where ¯aji denotes the complex conjugate of the scalar aji∈F. In particular, if A∈Rm×n,then note that AT=A∗.
Example 12.5.1:
With notation as in Example A.1.3,
AT=[3−11],BT=[40−12],CT=[142],DT=[1−13502214],ET=[6−14111323].
One of the motivations for defining the operations of transpose and conjugate transpose is that they interact with the usual arithmetic operations on matrices in a natural manner. We summarize the most fundamental of these interactions in the following theorem.
Theorem A.5.2. Given positive integers m,n∈Z+ and any matrices A,B∈Fm×n,
- (AT)T=A and (A∗)∗=A.
- (A+B)T=AT+BT and (A+B)∗=A∗+B∗.
- (αA)T=αAT and (αA)∗=αA∗, where α∈F is any scalar.
- (AB)T=BTAT.
- if m=n and A∈GL(n,F), then AT,A∗∈GL(n,F) with respective inverses given by
(AT)−1=(A−1)T and (A∗)−1=(A−1)∗.
Another motivation for defining the transpose and conjugate transpose operations is that they allow us to define several very special classes of matrices.
Definition A.5.3. Given a positive integer n∈Z+ , we say that the square matrix A∈Fn×n
- is symmetric if A=AT.
- is Hermitian if A=A∗.
- is orthogonal if A∈GL(n,R) and A−1=AT. Moreover, we define the (real) orthogonal group to be the set O(n)={A∈GL(n,R) | A−1=AT}.
- is unitary if A∈GL(n,C) and A−1=A∗. Moreover, we define the (complex)
unitary group to be the set U(n)={A∈GL(n,C) | A−1=A∗}.
A lot can be said about these classes of matrices. Both O(n) and U(n), for example, form a group under matrix multiplication. Additionally, real symmetric and complex Hermitian matrices always have real eigenvalues. Moreover, given any matrix A∈Rm×n,AAT is a symmetric matrix with real, non-negative eigenvalues. Similarly, for A∈Cm×n,AA∗ is Hermitian with real, non-negative eigenvalues.
A.5.2 The trace of a square matrix
Given a positive integer n∈Z+ and any square matrix A=(aij)∈Fn×n, we define the trace of A to be the scalar
trace(A)=n∑k=1akk∈F.
Example 12.5.2
With notation as in Example A.1.3 above,
trace(B)=4+2=6,trace(D)=1+0+4=5, and trace(E)=6+1+3=10.
Note, in particular, that the traces of A and C are not defined since these are not square matrices.
We summarize some of the most basic properties of the trace operation in the following theorem, including its connection to the transpose operations defined in the previous section.
Theorem A.5.5. Given positive integers m,n∈Z+ and square matrices A,B∈Fn×n,
- trace(αA)=αtrace(A), for any scalar α∈F.
- trace(A+B)=trace(A)+trace(B).
- trace(AT)=trace(A) and trace(A∗)=¯trace(A).
- trace(AA∗)=∑nk=1∑nl=1|akl|2. In particular, trace(AA∗)=0 if and only if A=0n×n .
- trace(AB)=trace(BA). More generally, given matrices A1,…,Am∈Fn×n, the trace operation has the so-called cyclic property, meaning that trace(A1⋯Am)=trace(A2⋯AmA1)=⋯=trace(AmA1⋯Am−1).
Moreover, if we define a linear map T:Fn→Fn by setting T(v)=Av for each v∈Fn and if T has distinct eigenvalues λ1,…,λn, then trace(A)=∑nk=1λk.
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
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