# 矩阵求导数

A

=

1

2

1

2

1

3

,

f

(

x

)

=

A

x

2

x

2

122113

,计算f(x)=∣∣x2∣∣Ax2的最大值。

f

(

x

)

=

x

1

x

2

x

3

1

2

2

1

1

3

1

2

1

2

1

3

x

1

x

2

x

3

x

1

2

+

x

2

2

+

x

3

2

=

x

1

x

2

x

3

5

0

7

0

5

1

7

1

10

x

1

x

2

x

3

x

1

2

+

x

2

2

+

x

3

2

f(x) = frac{ begin {vmatrix} x_1 x_2 x_3 end {vmatrix} begin {vmatrix} 1 & 2 \2 & -1 \1 & 3 end {vmatrix}begin {vmatrix} 1 & 2 & 1\2 & -1 & 3 end {vmatrix} begin {vmatrix} x_1\ x_2\x_3 end {vmatrix} }{x_1^2 + x_2^2 + x_3^2} =\ frac{ begin {vmatrix} x_1 x_2 x_3 end {vmatrix} begin {vmatrix} 5 & 0 & 7\0 & 5 & -1 \7 & -1 & 10end {vmatrix} begin {vmatrix} x_1\ x_2\x_3 end {vmatrix} }{x_1^2 + x_2^2 + x_3^2}

f(x)=x12+x22+x32

x1x2x3

121213

122113

x1x2x3

=x12+x22+x32

x1x2x3

5070517110

x1x2x3

x

T

A

T

A

x

x^T A ^ T A x

xTATAx求导:

(

x

+

Δ

x

)

T

A

T

A

(

x

+

Δ

x

)

=

(

x

T

+

Δ

x

T

)

A

T

A

(

x

+

Δ

x

)

=

(

x

T

A

T

A

+

Δ

x

T

A

T

A

)

(

x

+

Δ

x

)

=

x

T

A

T

A

x

+

x

T

A

T

A

Δ

x

+

Δ

x

T

A

T

A

x

+

Δ

x

T

A

T

A

Δ

x

(x + Delta x)^T A ^ T A (x + Delta x) =(x^T + Delta x ^ T) A ^ T A (x + Delta x) =\ (x^T A ^ T A + Delta x ^ T A ^ T A ) (x + Delta x) = \ x^T A ^ T A x +x^T A ^ T A Delta x + Delta x ^ T A ^ T A x + Delta x ^ T A ^ T A Delta x

(x+Δx)TATA(x+Δx)=(xT+ΔxT)ATA(x+Δx)=(xTATA+ΔxTATA)(x+Δx)=xTATAx+xTATAΔx+ΔxTATAx+ΔxTATAΔx

color{red}注意：上式中每一项都是一个标量，而不是向量。

(

x

+

Δ

x

)

T

A

T

A

(

x

+

Δ

x

)

x

T

A

T

A

x

Δ

x

=

x

T

A

T

A

Δ

x

+

Δ

x

T

A

T

A

x

+

Δ

x

T

A

T

A

Δ

x

Δ

x

=

2

A

T

A

x

frac { ||(x + Delta x)^T A ^ T A (x + Delta x) -x^T A ^ T A x ||}{ || Delta x||} = \ frac {||x^T A ^ T A Delta x + Delta x ^ T A ^ T A x + Delta x ^ T A ^ T A Delta x ||}{ ||Delta x ||} = 2 A^ TA x

∣∣Δx∣∣∣∣(x+Δx)TATA(x+Δx)xTATAx∣∣=∣∣Δx∣∣∣∣xTATAΔx+ΔxTATAx+ΔxTATAΔx∣∣=2ATAx

2

A

T

A

x

x

T

A

T

A

Δ

x

color{red}注意：上式取2 A^ TA x而不是x^ TA ^T A的原因是，\ Delta x一般是列向量，需要放在后面，此时需要将矩阵导数放在前面，并转置。

2

A

T

A

x

=

0

2 A^ TA x = 0

2ATAx=0 得：

?

color{red}此方法对吗?

THE END

)">