Applied Maths Al Cheat Sheet (214)
Determinants
b
Properties
: a bec
: =
·
det (I) =
1
(AB) det (B) 3
booa
a
det (A) * :
·
det
=
.
if det(A) A
0 is
singular
·
=
,
" "
A -
LU
a b c
-9 . _
+
9 +
·
det/n) =
ode =
product of
diagonals
alefl-bl if cla
0 Of or +
dez
(Eij) (2) 1
00i
det
·
= =
·
det (A) =
det(LU) =
det (L) det (U) =
det (n)
permutations
Pizz (Pijk) I l
det
I a
row swap Ca
·
=
,
singularity
·
if A has zero row , det(A) =
0
·
If one row of A is a multiple of another now
,
det (A) =
0
A =
zU
det (A) =
detCU) =
0
·
dez (AP) =
det (A)
everything about rows
applies to col's
Al =
X
Eigenvalues ('stretch factors) only lengthens from pre-mult by A
if & is evec of A
tr(A) As I trace =
sum of diagonals)
=
Acc =
X
(A-XI) #
=
0 ( must be a basis for mull space of A-XI)
↳
only non-mull of if A-XI is
singular
, Want det/A-XI) =
0 (singual
-
characteristic
nth
degree polynomidh >
polynomial
-
of M
eigenvalues are roots
* -
tr (A) x + det(A) =
0
↓ ↓
Levals product evals
Eigenvectors : 2 Where (A-xI) = 0 Lie
.
Finding well space)
for each eval Xi ↳ use LU decomp
: UC =
0
Eigenvalue Decomposition
AlP =
X ,
24 , Al =
XuEz Acts =
XsEs
,
[Abs As Alpas I
,
=
[Xx , xx , xxc ]
I di itif i I
kai) e
A i
-
f
As =
S-
1
A S-S
-
=
Diagonal matrices 1
%08] =(i) "T
:
S"S-
"
A =
Eigenvalues Properties :
·
A is square
I A -XII
·
=
O
2x2 : = X- tr (A) + det (A) =
(X -
X(X -
B) =
x -
( + B)x + <B
Determinants
b
Properties
: a bec
: =
·
det (I) =
1
(AB) det (B) 3
booa
a
det (A) * :
·
det
=
.
if det(A) A
0 is
singular
·
=
,
" "
A -
LU
a b c
-9 . _
+
9 +
·
det/n) =
ode =
product of
diagonals
alefl-bl if cla
0 Of or +
dez
(Eij) (2) 1
00i
det
·
= =
·
det (A) =
det(LU) =
det (L) det (U) =
det (n)
permutations
Pizz (Pijk) I l
det
I a
row swap Ca
·
=
,
singularity
·
if A has zero row , det(A) =
0
·
If one row of A is a multiple of another now
,
det (A) =
0
A =
zU
det (A) =
detCU) =
0
·
dez (AP) =
det (A)
everything about rows
applies to col's
Al =
X
Eigenvalues ('stretch factors) only lengthens from pre-mult by A
if & is evec of A
tr(A) As I trace =
sum of diagonals)
=
Acc =
X
(A-XI) #
=
0 ( must be a basis for mull space of A-XI)
↳
only non-mull of if A-XI is
singular
, Want det/A-XI) =
0 (singual
-
characteristic
nth
degree polynomidh >
polynomial
-
of M
eigenvalues are roots
* -
tr (A) x + det(A) =
0
↓ ↓
Levals product evals
Eigenvectors : 2 Where (A-xI) = 0 Lie
.
Finding well space)
for each eval Xi ↳ use LU decomp
: UC =
0
Eigenvalue Decomposition
AlP =
X ,
24 , Al =
XuEz Acts =
XsEs
,
[Abs As Alpas I
,
=
[Xx , xx , xxc ]
I di itif i I
kai) e
A i
-
f
As =
S-
1
A S-S
-
=
Diagonal matrices 1
%08] =(i) "T
:
S"S-
"
A =
Eigenvalues Properties :
·
A is square
I A -XII
·
=
O
2x2 : = X- tr (A) + det (A) =
(X -
X(X -
B) =
x -
( + B)x + <B
