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Exponents
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Nach A, C, B, a, b auflösen
b=C
b
=
C
Lösen
A = 0, C = 0, a = 0, B = 0<br/>A = -1, C = 0, a = 0, B = 1
A
=
0
,
C
=
0
,
a
=
0
,
B
=
0
A
=
−
1
,
C
=
0
,
a
=
0
,
B
=
1
Quiz
Complex Number
5 ähnliche Probleme wie:
\left. \begin{array} { l } { A ^ {C} = B }\\ { a = C }\\ { \text{Solve for } b \text{ where} } \\ { b = a } \end{array} \right.
A
C
=
B
a
=
C
Solve for
b
where
b
=
a
Ähnliche Aufgaben aus Websuche
Relationship between eigenvalues of A^TA and B=\left[\begin{array}{cc} 0 & A^T\\ A & 0\end{array}\right]
Relationship between eigenvalues of
A
T
A
and
B
=
[
0
A
A
T
0
]
https://math.stackexchange.com/questions/2014959/relationship-between-eigenvalues-of-ata-and-b-left-beginarraycc-0-a
Since \mathrm A^{\top} \mathrm A is positive semidefinite, its eigenvalues, \mu_1, \dots, \mu_n, are nonnegative. The characteristic polynomial of \mathrm B is \det (s \mathrm I_{2n} - \mathrm B) = \det \begin{bmatrix} s \mathrm I_{n} & -\mathrm A^{\top}\\ -\mathrm A & s \mathrm I_{n} \end{bmatrix} = \det (s^2 \mathrm I_{n} - \mathrm A^{\top} \mathrm A) ...
Since
A
⊤
A
is positive semidefinite, its eigenvalues,
μ
1
,
…
,
μ
n
, are nonnegative. The characteristic polynomial of
B
is
det
(
s
I
2
n
−
B
)
=
det
[
s
I
n
−
A
−
A
⊤
s
I
n
]
=
det
(
s
2
I
n
−
A
⊤
A
)
...
Sufficient condition for the block matrix \big(\begin{smallmatrix} B & A^T \\ A & 0 \end{smallmatrix} \big) to be invertible
Sufficient condition for the block matrix
(
B
A
A
T
0
)
to be invertible
https://math.stackexchange.com/questions/1027138/sufficient-condition-for-the-block-matrix-big-beginsmallmatrix-b-at-a
Note that \ker A = {\cal R}Z. Suppose Bu + A^T v =0, Au = 0. Then u \in \ker A= {\cal R} Z, hence u = Zw for some w. Then Z^T B Zw + Z^T A^T v = Z^T B Zw + (AZ)^T v = Z^T B Zw = 0. Hence w=0 ...
Note that
ker
A
=
R
Z
. Suppose
B
u
+
A
T
v
=
0
,
A
u
=
0
. Then
u
∈
ker
A
=
R
Z
, hence
u
=
Z
w
for some
w
. Then
Z
T
B
Z
w
+
Z
T
A
T
v
=
Z
T
B
Z
w
+
(
A
Z
)
T
v
=
Z
T
B
Z
w
=
0
. Hence
w
=
0
...
Cholesky factor when adding a row and column to already factorized matrix
Cholesky factor when adding a row and column to already factorized matrix
https://math.stackexchange.com/q/955874
To make this work, you should explicitly assume that a^tA^{-1}a<1; otherwise, the bordered matrix B:=\begin{bmatrix}A & a\\a^t & 1\end{bmatrix} would not be positive definite anymore (and ...
To make this work, you should explicitly assume that
a
t
A
−
1
a
<
1
; otherwise, the bordered matrix
B
:
=
[
A
a
t
a
1
]
would not be positive definite anymore (and ...
inverse of 2\times2 block matrix
inverse of
2
×
2
block matrix
https://math.stackexchange.com/q/1678502
One of the formulas here does not involve D^{-1}
One of the formulas here does not involve
D
−
1
Solve the Non-Homogeneous System y'=Cy+b(t)
Solve the Non-Homogeneous System
y
′
=
C
y
+
b
(
t
)
https://math.stackexchange.com/q/2801318
Let V= \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} and note that C = V \begin{bmatrix} -1 & 0 \\ 0 & -3 \end{bmatrix} V^{-1} and so e^{Ct} = V \begin{bmatrix} e^{-t} & 0 \\ 0 & e^{-3t} \end{bmatrix} V^{-1} ...
Let
V
=
[
1
1
1
−
1
]
and note that
C
=
V
[
−
1
0
0
−
3
]
V
−
1
and so
e
C
t
=
V
[
e
−
t
0
0
e
−
3
t
]
V
−
1
...
Solving systems of ODE'S in the form \dot{\overrightarrow{u}}=A\overrightarrow{u}+\overrightarrow{b}
Solving systems of ODE'S in the form
u
˙
=
A
u
+
b
https://math.stackexchange.com/questions/2034180/solving-systems-of-odes-in-the-form-dot-overrightarrowu-a-overrightarrow
You can use the method of undetermined coefficients or educated guessing to find a particular solution. Since 0 is not an eigenvalue of A and \vec b is constant, look for a particular solution ...
You can use the method of undetermined coefficients or educated guessing to find a particular solution. Since
0
is not an eigenvalue of
A
and
b
is constant, look for a particular solution ...
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Beispiele
Quadratische Gleichung
{ x } ^ { 2 } - 4 x - 5 = 0
x
2
−
4
x
−
5
=
0
Trigonometrie
4 \sin \theta \cos \theta = 2 \sin \theta
4
sin
θ
cos
θ
=
2
sin
θ
Lineare Gleichung
y = 3x + 4
y
=
3
x
+
4
Arithmetisch
699 * 533
6
9
9
∗
5
3
3
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
[
2
5
3
4
]
[
2
−
1
0
1
3
5
]
Simultane Gleichung
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
{
8
x
+
2
y
=
4
6
7
x
+
3
y
=
4
7
Differenzierung
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
d
x
d
(
x
−
5
)
(
3
x
2
−
2
)
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
∫
0
1
x
e
−
x
2
d
x
Grenzwerte
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}
x
→
−
3
lim
x
2
+
2
x
−
3
x
2
−
9
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