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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,nZ+ 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 ajiF. In particular, if ARm×n,then note that AT=A.

Example 12.5.1:

With notation as in Example A.1.3,

AT=[311],BT=[4012],CT=[142],DT=[113502214],ET=[614111323].

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,nZ+ and any matrices A,BFm×n,

  1. (AT)T=A and (A)=A.
  2. (A+B)T=AT+BT and (A+B)=A+B.
  3. (αA)T=αAT and (αA)=αA, where αF is any scalar.
  4. (AB)T=BTAT.
  5. if m=n and AGL(n,F), then AT,AGL(n,F) with respective inverses given by

(AT)1=(A1)T and (A)1=(A1).

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 nZ+ , we say that the square matrix AFn×n

  1. is symmetric if A=AT.
  2. is Hermitian if A=A.
  3. is orthogonal if AGL(n,R) and A1=AT. Moreover, we define the (real) orthogonal group to be the set O(n)={AGL(n,R) | A1=AT}.
  4. is unitary if AGL(n,C) and A1=A. Moreover, we define the (complex)

unitary group to be the set U(n)={AGL(n,C) | A1=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 ARm×n,AAT is a symmetric matrix with real, non-negative eigenvalues. Similarly, for ACm×n,AA is Hermitian with real, non-negative eigenvalues.

A.5.2 The trace of a square matrix

Given a positive integer nZ+ and any square matrix A=(aij)Fn×n, we define the trace of A to be the scalar

trace(A)=nk=1akkF.

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,nZ+ and square matrices A,BFn×n,

  1. trace(αA)=αtrace(A), for any scalar αF.
  2. trace(A+B)=trace(A)+trace(B).
  3. trace(AT)=trace(A) and trace(A)=¯trace(A).
  4. trace(AA)=nk=1nl=1|akl|2. In particular, trace(AA)=0 if and only if A=0n×n .
  5. trace(AB)=trace(BA). More generally, given matrices A1,,AmFn×n, the trace operation has the so-called cyclic property, meaning that trace(A1Am)=trace(A2AmA1)==trace(AmA1Am1).

Moreover, if we define a linear map T:FnFn by setting T(v)=Av for each vFn and if T has distinct eigenvalues λ1,,λn, then trace(A)=nk=1λk.


This page titled 12.5: Special operations on matrices is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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