orthogonal_matrix ( A ) : = ( A ∈ R n × n and A T A = I n ) \text{orthogonal\_matrix}(A):=\left(A\in \R^{n\times n}\;\text{and}\;A^TA=I_n\right) orthogonal_matrix(A):=(A∈Rn×nandATA=In)
diagonal_matrix
(
A
)
:
=
A
∈
R
n
×
n
and
∀
i
∈
{
1
,
⋯
,
n
}
∀
j
∈
{
1
,
⋯
,
n
}
(
A
i
j
≠
0
⇒
i
=
j
)
diag
(
λ
1
,
⋯
,
λ
n
)
:
=
A
where
A
∈
R
n
×
n
and
(
∀
i
∈
{
1
,
⋯
,
n
}
A
i
i
=
λ
i
)
and
diagonal_matrix
(
A
)
symmetric_matrix ( A ) : = ( A ∈ R n × n and A T = A ) \text{symmetric\_matrix}(A):=\left(A\in \R^{n\times n} \; \text{and}\; A^T=A\right) symmetric_matrix(A):=(A∈Rn×nandAT=A)
quadratic_form
A
:
=
f
where
f
:
R
n
↦
R
,
f
(
x
)
=
x
T
A
x
A ∼ B : = ∃ P orthogonal_matrix ( P ) and A = P T B P A\sim B:=\exist P\;\text{orthogonal\_matrix}(P)\text{\;and\;}A=P^TBP A∼B:=∃Porthogonal_matrix(P)andA=PTBP
eigenvalue_set ( A ) : = { λ ∣ ∃ x ∈ R n ( x ≠ 0 and A x = λ x ) } \text{eigenvalue\_set}(A):=\{\lambda\;|\;\exist x\in R^n\;(x\neq0\text{\;and\;}Ax=\lambda x)\} eigenvalue_set(A):={λ∣∃x∈Rn(x=0andAx=λx)}
∀
A
symmetric_matrix
(
A
)
⇒
∃
Λ
∃
P
(
diagonal_matrix
(
Λ
)
and
orthogonal_matrix
(
P
)
and
A
∼
Λ
and
{
Λ
i
i
∣
i
∈
{
1
,
⋯
,
n
}
}
=
eigenvalue_set
(
A
)
)
symmetric_matrix ( A ) ⇒ max eigenvalue_set ( A ) = max x ∈ R n , ∥ x ∥ = 1 { quadratic_form A ( x ) } \text{symmetric\_matrix}(A) \Rightarrow \text{max}\;\text{eigenvalue\_set}(A)=\max_{x\in\R^n,\|x\|=1}\{\text{quadratic\_form}_A(x)\} symmetric_matrix(A)⇒maxeigenvalue_set(A)=x∈Rn,∥x∥=1max{quadratic_formA(x)}