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❋❋❚ Pr♦❣r❛♠ ●❡♥❡r❛t✐♦♥ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r②✿ ❙▼P ❛♥❞ ▼✉❧t✐❝♦r❡ ❋✳ ❋r❛♥❝❤❡tt✐✱ ❨✳ ❱♦r♦♥❡♥❦♦✱ ▼✳ Püs❝❤❡❧ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣ ❈❛r♥❡❣✐❡ ▼❡❧❧♦♥ ❯♥✐✈❡rs✐t② ◆✐❝♦❧❛ ▼❛r❝❛❝❝✐ ❘♦ss✐✱ ✸✶✳ ❖❦t♦❜❡r ✷✵✶✶

❖✈❡r✈✐❡✇ � ❋✐rst P❛rt✿ ❋r♦♠ t❤❡ ❛❧❣♦r✐t❤♠ t♦ t❤❡ ❝♦❞❡ � ❙❡❝♦♥❞ P❛rt✿ ❙♣✐r❛❧ ❡①t❡♥s✐♦♥ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r② � ❉✐s❝✉ss✐♦♥

❆❧❣♦r✐t❤♠s✿ ❉✐r❡❝t ▼❛tr✐①✲❱❡❝t♦r ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✿ ❈♦♦❧❡②✲❚✉❦❡② ❋❋❚ ❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ▼❛tr✐① ❢❛❝t♦r✐③❛t✐♦♥✿ ❉❋❚ ❉❋❚ ❉❋❚ ❆ss✉♠❡ ❇❛s❡ ❝❛s❡✿ ❉❋❚ ❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ � ❚❤❡ ♣r♦❜❧❡♠✿ x ∈ C N ●✐✈❡♥✿ y = ❉❋❚ N x ❈♦♠♣✉t❡✿ ❉❋❚ N = [ w kl N ] 0 ≤ k,l<N w n = e − 2 πi/N

❈♦♦❧❡②✲❚✉❦❡② ❋❋❚ ❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ▼❛tr✐① ❢❛❝t♦r✐③❛t✐♦♥✿ ❉❋❚ ❉❋❚ ❉❋❚ ❆ss✉♠❡ ❇❛s❡ ❝❛s❡✿ ❉❋❚ ❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ � ❆❧❣♦r✐t❤♠s✿ � ❚❤❡ ♣r♦❜❧❡♠✿ x ∈ C N ◮ ❉✐r❡❝t ▼❛tr✐①✲❱❡❝t♦r ●✐✈❡♥✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ O ( N 2 ) y = ❉❋❚ N x ❈♦♠♣✉t❡✿ ◮ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✿ ❉❋❚ N = [ w kl N ] 0 ≤ k,l<N O ( N log N ) w n = e − 2 πi/N

❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ � ❆❧❣♦r✐t❤♠s✿ � ❚❤❡ ♣r♦❜❧❡♠✿ x ∈ C N ◮ ❉✐r❡❝t ▼❛tr✐①✲❱❡❝t♦r ●✐✈❡♥✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ O ( N 2 ) y = ❉❋❚ N x ❈♦♠♣✉t❡✿ ◮ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✿ ❉❋❚ N = [ w kl N ] 0 ≤ k,l<N O ( N log N ) w n = e − 2 πi/N � ❈♦♦❧❡②✲❚✉❦❡② ❋❋❚ ◮ ❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ◮ ▼❛tr✐① ❢❛❝t♦r✐③❛t✐♦♥✿ ❉❋❚ mn x = ( ❉❋❚ m ⊗ I n ) D m,n ( I m ⊗ ❉❋❚ n ) L mn m x ◮ ❆ss✉♠❡ N = 2 k ◮ ❇❛s❡ ❝❛s❡✿ ❉❋❚ 2

❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ � ❆❧❣♦r✐t❤♠s✿ � ❚❤❡ ♣r♦❜❧❡♠✿ x ∈ C N ◮ ❉✐r❡❝t ▼❛tr✐①✲❱❡❝t♦r ●✐✈❡♥✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ O ( N 2 ) y = ❉❋❚ N x ❈♦♠♣✉t❡✿ ◮ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✿ ❉❋❚ N = [ w kl N ] 0 ≤ k,l<N O ( N log N ) w n = e − 2 πi/N � ❈♦♦❧❡②✲❚✉❦❡② ❋❋❚ ◮ ❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ◮ ▼❛tr✐① ❢❛❝t♦r✐③❛t✐♦♥✿ ❉❋❚ mn x = ( ❉❋❚ m ⊗ I n ) D m,n ( I m ⊗ ❉❋❚ n ) L mn m x ◮ ❆ss✉♠❡ N = 2 k ◮ ❇❛s❡ ❝❛s❡✿ ❉❋❚ 2

❙♣✐r❛❧✿ ❣❡♥❡r❛t✐♥❣ ♦♣t✐♠✐③❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s � ❙t❛rt ❢♦r♠✉❧❛✿ ❉❋❚ N � ❈♦♦❧❡②✲❚✉❦❡② ❘✉❧❡✿ ❉❋❚ mn → ( ❉❋❚ m ⊗ In ) D m,n ( I m ⊗ ❉❋❚ n ) L mn m ✶ ✶s♦✉r❝❡✿ ❋❋❚ Pr♦❣r❛♠ ●❡♥❡r❛t✐♦♥ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r②✿ ❙▼P ❛♥❞ ▼✉❧t✐❝♦r❡

❙♣✐r❛❧✿ ❣❡♥❡r❛t✐♥❣ ♦♣t✐♠✐③❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s � ❙t❛rt ❢♦r♠✉❧❛✿ ❉❋❚ N � ❈♦♦❧❡②✲❚✉❦❡② ❘✉❧❡✿ ❉❋❚ mn → ( ❉❋❚ m ⊗ In ) D m,n ( I m ⊗ ❉❋❚ n ) L mn m � ❊①❛♠♣❧❡✿ ❉❋❚ 8 = ( ❉❋❚ 2 ⊗ I 4 ) D 8 , 4 ( I 2 ⊗ ( ❉❋❚ 2 ⊗ I 2 ) D 4 , 2 ( I 2 ⊗ ❉❋❚ 2 ) L 4 2 ) L 9 2 ✶ ✶s♦✉r❝❡✿ ❋❋❚ Pr♦❣r❛♠ ●❡♥❡r❛t✐♦♥ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r②✿ ❙▼P ❛♥❞ ▼✉❧t✐❝♦r❡

❙♣✐r❛❧✿ ❣❡♥❡r❛t✐♥❣ ♦♣t✐♠✐③❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s � ❙t❛rt ❢♦r♠✉❧❛✿ ❉❋❚ N � ❈♦♦❧❡②✲❚✉❦❡② ❘✉❧❡✿ ❉❋❚ mn → ( ❉❋❚ m ⊗ In ) D m,n ( I m ⊗ ❉❋❚ n ) L mn m � ❙P▲ t♦ ❝♦❞❡ tr❛♥s❧❛t✐♦♥ t❛❜❧❡✿ y = ( A n B n ) x t❬✵✿✶✿♥✲✶❪ ❂ ❇✭①❬✵✿✶✿♥✲✶❪✮❀ ②❬✵✿✶✿♥✲✶❪ ❂ ❆✭t✭❬✵✿✶✿♥✲✶❪✮❀ y = ( I m ⊗ A n ) x ❢♦r ✭✐❂✵❀✐❁♠❀✐✰✰✮ ②❬✐✯♥✿✶✿✐✯♥✰♥✲✶❪ ❂ ❆✭①❬✐✯♥✿✶✿✐✯♥✰♥✲✶❪✮❀ ✳✳✳ ✳✳✳ ✶ ✶s♦✉r❝❡✿ ❋❋❚ Pr♦❣r❛♠ ●❡♥❡r❛t✐♦♥ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r②✿ ❙▼P ❛♥❞ ▼✉❧t✐❝♦r❡

❙♣✐r❛❧✿ ❣❡♥❡r❛t✐♥❣ ♦♣t✐♠✐③❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s � ❙t❛rt ❢♦r♠✉❧❛✿ ❉❋❚ N � ❈♦♦❧❡②✲❚✉❦❡② ❘✉❧❡✿ ❉❋❚ mn → ( ❉❋❚ m ⊗ In ) D m,n ( I m ⊗ ❉❋❚ n ) L mn m � ❙P▲ t♦ ❝♦❞❡ tr❛♥s❧❛t✐♦♥ t❛❜❧❡✿ y = ( A n B n ) x t❬✵✿✶✿♥✲✶❪ ❂ ❇✭①❬✵✿✶✿♥✲✶❪✮❀ ②❬✵✿✶✿♥✲✶❪ ❂ ❆✭t✭❬✵✿✶✿♥✲✶❪✮❀ y = ( I m ⊗ A n ) x ❢♦r ✭✐❂✵❀✐❁♠❀✐✰✰✮ ②❬✐✯♥✿✶✿✐✯♥✰♥✲✶❪ ❂ ❆✭①❬✐✯♥✿✶✿✐✯♥✰♥✲✶❪✮❀ ✳✳✳ ✳✳✳ ✶ � ❙❡❛r❝❤ s♣❛❝❡✿ ❢❛❝t♦r✐③❛t✐♦♥s ❛♥❞ ❜❛s❡ ❝❛s❡s ✶s♦✉r❝❡✿ ❋❋❚ Pr♦❣r❛♠ ●❡♥❡r❛t✐♦♥ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r②✿ ❙▼P ❛♥❞ ▼✉❧t✐❝♦r❡

❙t❡♣s✿ ■❞❡♥t✐❢② ♣❛r❛❧❧❡❧ ❝♦♥str✉❝ts ✭❛♥❞ t❤❡✐r ✐♠♣❧❡♠❡♥t❛t✐♦♥✮ ❉❡✜♥❡ r❡✇r✐t✐♥❣ r✉❧❡s ❉❡r✐✈❡ ❛ ♣❛r❛❧❧❡❧ ❈♦♦❧❡②✲❚✉❦❡② ■ss✉❡s✿ ▲♦❛❞ ❇❛❧❛♥❝✐♥❣ ❙②♥❝❤r♦♥✐③❛t✐♦♥ ♦✈❡r❤❡❛❞ ❋❛❧s❡ ❙❤❛r✐♥❣ ❊①t❡♥❞✐♥❣ ❙♣✐r❛❧ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r② � ◆❡✇ t❛❣✿ ❉❋❚ N � �� � s♠♣ ( p,µ ) ◮ ◆✉♠❜❡r ♦❢ ♣r♦❝❡ss♦rs✿ p ◮ ❈❛❝❤❡ ❧✐♥❡ s✐③❡✿ µ � ❋✐♥❞ ♣❛r❛❧❧❡❧ ❈♦♦❧❡②✲❚✉❦❡②✿ ❉❋❚ mn → · · · � �� � s♠♣ ( p,µ )

■ss✉❡s✿ ▲♦❛❞ ❇❛❧❛♥❝✐♥❣ ❙②♥❝❤r♦♥✐③❛t✐♦♥ ♦✈❡r❤❡❛❞ ❋❛❧s❡ ❙❤❛r✐♥❣ ❊①t❡♥❞✐♥❣ ❙♣✐r❛❧ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r② � ❙t❡♣s✿ � ◆❡✇ t❛❣✿ ◮ ■❞❡♥t✐❢② ♣❛r❛❧❧❡❧ ❝♦♥str✉❝ts ❉❋❚ N � �� � ✭❛♥❞ t❤❡✐r ✐♠♣❧❡♠❡♥t❛t✐♦♥✮ s♠♣ ( p,µ ) ◮ ❉❡✜♥❡ r❡✇r✐t✐♥❣ r✉❧❡s ◮ ❉❡r✐✈❡ ❛ ♣❛r❛❧❧❡❧ ❈♦♦❧❡②✲❚✉❦❡② ◮ ◆✉♠❜❡r ♦❢ ♣r♦❝❡ss♦rs✿ p ◮ ❈❛❝❤❡ ❧✐♥❡ s✐③❡✿ µ � ❋✐♥❞ ♣❛r❛❧❧❡❧ ❈♦♦❧❡②✲❚✉❦❡②✿ ❉❋❚ mn → · · · � �� � s♠♣ ( p,µ )

❊①t❡♥❞✐♥❣ ❙♣✐r❛❧ ❢♦r ❙❤❛r❡❞ ▼❡♠♦r② � ❙t❡♣s✿ � ◆❡✇ t❛❣✿ ◮ ■❞❡♥t✐❢② ♣❛r❛❧❧❡❧ ❝♦♥str✉❝ts ❉❋❚ N � �� � ✭❛♥❞ t❤❡✐r ✐♠♣❧❡♠❡♥t❛t✐♦♥✮ s♠♣ ( p,µ ) ◮ ❉❡✜♥❡ r❡✇r✐t✐♥❣ r✉❧❡s ◮ ❉❡r✐✈❡ ❛ ♣❛r❛❧❧❡❧ ❈♦♦❧❡②✲❚✉❦❡② ◮ ◆✉♠❜❡r ♦❢ ♣r♦❝❡ss♦rs✿ p ◮ ❈❛❝❤❡ ❧✐♥❡ s✐③❡✿ µ � ■ss✉❡s✿ ◮ ▲♦❛❞ ❇❛❧❛♥❝✐♥❣ � ❋✐♥❞ ♣❛r❛❧❧❡❧ ❈♦♦❧❡②✲❚✉❦❡②✿ ◮ ❙②♥❝❤r♦♥✐③❛t✐♦♥ ♦✈❡r❤❡❛❞ ❉❋❚ mn → · · · � �� � ◮ ❋❛❧s❡ ❙❤❛r✐♥❣ s♠♣ ( p,µ )

❋❛❧s❡ s❤❛r✐♥❣ � ❈✐r❝✉♠st❛♥❝❡s✿ � ▲❡❛❞s t♦ ❝❛❝❤❡ ❧✐♥❡ t❤r❛s❤✐♥❣ ◮ ❉✐✛❡r❡♥t ❞❛t❛ � ❙♦❧✉t✐♦♥✿ ♦♥❡ ♣r♦❝❡ss♦r ♣❡r ❝❛❝❤❡ ◮ ❙❛♠❡ ❝❛❝❤❡ ❧✐♥❡ ❧✐♥❡ ◮ ❈♦♥s❡❝✉t✐✈❡ ❛❝❝❡ss❡s