Processing math: 100%
Skip to main content
🔗

Section 39.2 Terminology and notation

🔗
We will make the following two definitions relative to the standard inner products on Rn and Cn simultaneously. So, when interpreting the definitions for the real case, take V=Rn and ⟨−,−⟩ to mean the dot product. And when interpreting the definitions for the complex case, take V=Cn and ⟨−,−⟩ to mean the complex dot product.

adjoint matrix (of an n×n matrix A)

the matrix A∗ that satisfies

⟨u,Av⟩=⟨A∗u,v⟩

for every pair of column vectors u,v in V

self-adjoint matrix

a matrix for which A∗=A, so that A satisfies

⟨u,Av⟩=⟨Au,v⟩

for every pair of column vectors u,v in V; usually called symmetric in the real context and Hermitian in the complex context

🔗
The following two terminology instances are already differentiated between the real and complex contexts, though they describe the same concept.

orthogonal matrix

a real matrix A that satisfies

⟨Au,Av⟩R=⟨u,v⟩R

for every pair of column vectors u,v in Rn

unitary matrix

a complex matrix A that satisfies

⟨Au,Av⟩C=⟨u,v⟩C

for every pair of column vectors u,v in Cn

🔗
For convenience, we will unify the two versions of the concept above into one terminology instance. Again, take V=Rn and ⟨−,−⟩ to mean the dot product in the real context, and take V=Cn and ⟨−,−⟩ to mean the complex dot product in the complex context.

product-preserving matrix

a matrix A that satisfies

⟨Au,Av⟩=⟨u,v⟩

for every pair of column vectors u,v in V