[Zhan Xiang matrix theory exercise reference] exercise 3.15

15. set $ s_n [a, B] $ to indicate that all elements belong to the set of real symmetric matrices of the given range $ [a, B] $. for $ j = 1, N $ OK $ \ Bex \ MAX \ sed {\ lm_j (a); \ A \ In s_n [, b]} \ mbox {And} \ min \ sed {\ lm_j (a); \ A \ In s_n [a, B]}, \ EEx $ and the matrix that obtains the maximum and minimum values respectively.

 

 

Answer: For $0 \ NEQ x \ In \ BBR ^ N $, $ \ beex \ Bea & \ quad x ^ tax \\& = x ^ TP ^ t (PAP ^ t) px \ & \ quad \ sex {P \ mbox {is a replacement array, so that} front of PX \ mbox {} k \ mbox {component} \ geq 0, \ mbox {post} k \ mbox {parts} <0; \ atop \ mbox {and may wish to set} k> 0, \ mbox {otherwise, use}-x \ mbox {to replace} X }\\& = y ^ tby \ quad \ sex {B = pap ^ t \ mbox {. The element is} \ mbox {elements of the rearrangement }, y = px }\\\& =\ sum _ {I, j = 1} ^ n B _ {IJ} y_iy_j \\\& =\ sum _ {I, j = 1} ^ k B _ {IJ} y_iy_j + 2 \ sum _ {I = 1} ^ k \ sum _ {J = k + 1} ^ n B _ {IJ} y_iy_j + \ sum _ {I, j = k + 1} ^ n B _ {IJ} y_iy_j \\& \ geq A \ sum _ {I, j = 1} ^ K y_iy_j + 2B \ sum _ {I = 1} ^ k \ sum _ {J = k + 1} ^ n y_iy_j + A \ sum _ {I, j = k + 1} ^ n y_ I y_j \ & = y ^ tjy \ & \ quad \ sex {J = \ sex {\ BA {CC} aj_k & BJ _{ k, n-k} \ BJ _ {n-k, k} & AJ _ {n-k} \ EA}, J _ {R, s} \ mbox {is} r \ times s \ mbox {level matrix} of each element} 1 \ mbox {, j_r = J _ {R, R }}. \ EEA \ eeex $ therefore, $ \ beex \ Bea \ lm_n () & =\ min _ {\ Sen {x} = 1} x ^ tax \\\\=\ min _ {\ Sen {y} = 1} y ^ * JY \\& = \ lm_n (j ). \ EEA \ eeex $ returns the smallest feature value of $ J $ \ lm_n (j) $. apparently, $ J $ can be converted to $ \ Bex \ sex {\ BA {cccccc} A & \ cdots & A & B & \ cdots & B \ 0 & \ cdots & 0 & 0 & \ cdots & 0 \ vdots & \ vdots \ 0 & \ cdots & 0 & \ cdots & 0 \ cdots & 0 \ \ B & \ cdots & B & A & \ cdots & A \ EA }, \ EEx $ rank $ \ Leq 2 $. After you convert $ J $ to a diagonal matrix, you can see that $ J $ has a maximum of $2 $ non-zero feature values, recorded as $ \ mu_1 $, $ \ mu_2 $, the Frobenius norm is passed through the comparison trace (the uniu is unchanged, but the orthogonal is unchanged ), $ \ Bex C \ equiv na =\mu_1 + \ mu_2, \ EEx $ \ Bex d \ equiv K ^ 2a ^ 2 + 2 k (n-k) B ^ 2 + (n-k) ^ 2a ^ 2 = \ mu_1 ^ 2 + \ mu_2 ^ 2. \ EEx $ and $ \ mu_1, \ mu_2 $ is the quadratic equation $ \ Bex t ^ 2-ct + \ frac {C ^ 2-D} {2} = 0 \ EEx $. so $ \ beex \ Bea \ lm_n (j) & =\ frac {C-\ SQRT {C ^ 2-4 \ frac {C ^ 2-D} {2 }}{ 2 }\\\\=\ frac {1} {2} \ SEZ {Na-\ SQRT {(n-2k) ^ 2a ^ 2 + 4 K (n-k) B ^ 2 }\\\=\ frac {1} {2} \ SEZ {Na-\ SQRT {4 (a ^ 2-B ^ 2) k ^ 2-4 (a ^ 2-B ^ 2) NK + N ^ 2a ^ 2 }}. \ EEA \ eeex $ therefore, when $ | A | <B $, if $ N $ is an even number, then, when and only when $ \ DPS {k = \ frac {n} {2 }}$, $ \ lm_n (j) $ is minimized, $ \ Bex \ frac {n (a-B)} {2}; \ EEx $ when $ N $ is an odd number, only when $ \ DPS {k = \ frac {n-1} {2 }$ or $ \ DPS {k = \ frac {n + 1} {2} $, $ \ lm_n (j) $ to reach the minimum, $ \ Bex \ frac {1} {2} \ SEZ {Na-\ SQRT {A ^ 2 + (n ^ 2-1) B ^ 2 }}. \ EEx $ when $ | A | = B $, $ A <0 $, for $ \ forall \ 1 \ Leq k \ Leq N $, $ \ Bex \ lm_n (j) = Na. \ EEx $ when $ | A |> B $, $ A <0 $. if and only when $ k = N $, $ \ lm_n (j) the minimum value is $ \ Bex Na. \ EEx $ to sum up, we will summarize the following:

 

(1 ). when $ | A | <B $, if $ N $ is an even number, only when $ A $ and $ \ Bex \ sex {\ BA {CC} AJ _ \ frac {n} {2} & BJ _ \ frac {n }{ 2} \ BJ _ \ frac {n} {2} & AJ _ \ frac {n} {2} \ EA} \ EEx $ replace similarity, $ \ lm_n (a) $ reaches the minimum value, which is $ \ Bex \ frac {n (a-B)} {2}; \ EEx $ if $ N $ is an odd number, only when $ A $ and $ \ Bex \ sex {\ BA {CC} AJ _ \ frac {n-1} {2} & BJ _ {\ frac {n-1} {2 }, \ frac {n + 1} {2 }}\\ BJ _ {\ frac {n + 1} {2 }, \ frac {n-1} {2 }}& AJ _ \ frac {n + 1} {2} \ EA} \ EEx $ replace similarity, $ \ lm_n () the minimum value is $ \ Bex \ frac {1} {2} \ SEZ {Na-\ SQRT {A ^ 2 + (n ^ 2-1) B ^ 2 }}; \ EEx $

 

(2 ). when $ | A | = B $, if and only if some $1 \ Leq k \ Leq N $ exists, $ A $ and $ \ Bex \ sex {\ BA {CC} aj_k & BJ _ {k, n-k} \ BJ _ {n-k, k} & AJ _ {n-k} \ EA} \ EEx $ when the replacement is similar, the minimum value of $ \ lm_n (a) $ is $ \ Bex Na. \ EEx $

 

(3 ). when $ | A |> B $, when and only when $ A = aj_n $, $ \ lm_n (a) $ reaches the minimum value, which is $ \ Bex Na. \ EEx $ last, $ \ Bex \ MAX \ sed {\ lm_1 (a); A \ In s_n [a, B]}, \ EEx $ \ Bex \ min \ sed {\ lm_1 (a); A \ In s_n [a, B]}, \ EEx $ \ Bex \ MAX \ sed {\ lm_n (a); A \ In s_n [, b]} \ EEx $ can be discussed in a similar way and the corresponding conclusions are obtained.

[Zhan Xiang matrix theory exercise reference] exercise 3.15