Contents Structured Rank Matrices 1 The nullity theorem Lecture 2: - PowerPoint PPT Presentation

Contents Structured Rank Matrices 1 The nullity theorem Lecture 2: The theorem Structure Transport Proofs Examples related to structured ranks References Marc Van Barel and Raf Vandebril Dept. of Computer Science, K.U.Leuven, Belgium Chemnitz, Germany, 26-30 September 2011 2 Generalizations of the nullity theorem The LU -decomposition The QR -decomposition References Structured Rank Matrices Lecture 2: Structure Transport 2 / 26 � The nullity theorem The nullity theorem Outline The nullity theorem Definition (Right null space) Given a matrix A ∈ R m × n . The right null space N ( A ) equals 1 The nullity theorem N ( A ) = { x ∈ R n | A x = 0 } . The theorem Proofs Definition (Nullity of a matrix) Examples related to structured ranks References Given a matrix A ∈ R m × n . The nullity n( A ) is defined as the dimension of the right null space of A . 2 Generalizations of the nullity theorem Corollary The LU -decomposition The dimension of the right null space corresponds to the rank deficiency The QR -decomposition of the columns of the matrix A: References n( A ) = n − rank ( A ) = (number of columns) − rank ( A ) . Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 3 / 26 4 / 26 � �

The nullity theorem The nullity theorem The nullity theorem Corollaries of the nullity theorem Corollary Theorem (Nullity theorem) Suppose A ∈ R n × n is a nonsingular matrix, and α, β are nonempty Suppose the following invertible matrix A ∈ R n × n is partitioned as subsets of N with | α | < n and | β | < n. Then � � A 11 A 12 � A − 1 ( α ; β ) � rank = rank ( A ( N \ β ; N \ α )) + | α | + | β | − n . A = A 21 A 22 with A 11 of size p × q. The inverse B of A is partitioned as Proof: Permuting the matrix such that A ( N \ β ; N \ α ) moves to the upper left � � B 11 B 12 A − 1 = B = position A 11 , will move A − 1 ( α ; β ) to the position B 22 . Using the B 21 B 22 equalities: with B 11 of size q × p. Then the nullities n( A 11 ) and n( B 22 ) are equal: n( A 11 ) = n − | α | − rank ( A 11 ) , n( B 22 ) = | β | − rank ( B 22 ) , n( A 11 ) = n( B 22 ) . gives us the proof. Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 4 / 26 5 / 26 � � The nullity theorem The nullity theorem Corollaries of the nullity theorem Corollaries of the nullity theorem Corollary Corollary Suppose A ∈ R n × n is a nonsingular matrix, and α, β are nonempty Suppose A ∈ R n × n is a nonsingular matrix, and α, β are nonempty subsets of N with | α | < n and | β | < n. Then subsets of N with | α | < n and | β | < n. Then A − 1 ( α ; β ) A − 1 ( α ; β ) � � � � rank = rank ( A ( N \ β ; N \ α )) + | α | + | β | − n . rank = rank ( A ( N \ β ; N \ α )) + | α | + | β | − n . Examples for 5 × 5 matrices: Examples for 5 × 5 matrices: α = { 1 , 2 } and N \ β = { 3 , 4 , 5 } and α = { 1 , 2 } and N \ β = { 4 , 5 } and β = { 1 , 2 } N \ α = { 3 , 4 , 5 } β = { 1 , 2 , 3 } N \ α = { 3 , 4 , 5 }         × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×                 × × × × × ↔ × × × × × × × × × × ↔ × × × × ×                 × × × × × × × × × × × × × × × × × × × ×         × × × × × × × × × × × × × × × × × × × × Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 5 / 26 5 / 26 � �

The nullity theorem The nullity theorem Corollaries of the nullity theorem Corollaries of the nullity theorem Corollary Corollary Suppose A ∈ R n × n is a nonsingular matrix, and α, β are nonempty Suppose A ∈ R n × n is a nonsingular matrix, and α, β are nonempty subsets of N with | α | < n and | β | < n. Then subsets of N with | α | < n and | β | < n. Then � A − 1 ( α ; β ) � � A − 1 ( α ; β ) � rank = rank ( A ( N \ β ; N \ α )) + | α | + | β | − n . rank = rank ( A ( N \ β ; N \ α )) + | α | + | β | − n . Examples for 5 × 5 matrices: Examples for 5 × 5 matrices: α = { 3 , 4 , 5 } and N \ β = { 3 , 4 , 5 } and α = { 2 , 4 } and N \ β = { 2 , 4 , 5 } and β = { 1 , 2 } N \ α = { 1 , 2 } β = { 1 , 3 } N \ α = { 1 , 3 , 5 }         × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×                 × × × × × ↔ × × × × × × × × × × ↔ × × × × ×                 × × × × × × × × × × × × × × × × × × × ×         × × × × × × × × × × × × × × × × × × × × Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 5 / 26 5 / 26 � � The nullity theorem The nullity theorem Some corollaries of the nullity theorem Some corollaries of the nullity theorem Corollary Corollary For a nonsingular matrix A ∈ R n × n and α ⊆ N, we have: For a nonsingular matrix A ∈ R n × n and α ⊆ N, we have: A − 1 ( α ; N \ α ) A − 1 ( α ; N \ α ) � � � � rank = rank ( A ( α ; N \ α )) . rank = rank ( A ( α ; N \ α )) . Proof: This means that for a matrix the following blocks always have the same rank in A and in A − 1 . Is a direct consequence of the previous equation: α = { 2 , 3 , 4 , 5 } and α = { 3 , 4 , 5 } and � � A − 1 ( α ; β ) rank = rank ( A ( N \ β ; N \ α )) + | α | + | β | − n , N \ α = { 1 } N \ α = { 1 , 2 }  × × × × ×   × × × × ×  when posing β = N \ α : × × × × × × × × × × � �     A − 1 ( α ; N \ α ) rank = rank ( A ( α ; N \ α )) + | α | + | N \ α | − n .     × × × × × × × × × ×         × × × × × × × × × ×     × × × × × × × × × × Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 6 / 26 6 / 26 � �

The nullity theorem The nullity theorem Some corollaries of the nullity theorem Some corollaries of the nullity theorem Corollary Corollary For a nonsingular matrix A ∈ R n × n and α ⊆ N, we have: For a nonsingular matrix A ∈ R n × n and α ⊆ N, we have: A − 1 ( α ; N \ α ) A − 1 ( α ; N \ α ) � � � � rank = rank ( A ( α ; N \ α )) . rank = rank ( A ( α ; N \ α )) . This means that for a matrix the following blocks always have the same This means that for a matrix the following blocks always have the same rank in A and in A − 1 . rank in A and in A − 1 . α = { 4 , 5 } and α = { 5 } and α = { 3 , 5 } and α = { 2 , 3 } and N \ α = { 1 , 2 , 3 } N \ α = { 1 , 2 , 3 , 4 } N \ α = { 1 , 2 , 4 } N \ α = { 1 , 4 , 5 }         × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×                 × × × × × × × × × × × × × × × × × × × ×                 × × × × × × × × × × × × × × × × × × × ×         × × × × × × × × × × × × × × × × × × × × Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 6 / 26 6 / 26 � � The nullity theorem The nullity theorem Outline Different proofs There exist different strategies to prove the nullity theorem. An important remark, the theorem predicts structures but does not provide inversion formulas. 1 The nullity theorem Fiedler and Markham proved it, working directly on the ranks and nullities The theorem of the blocks, their proof was based on a paper by Gustafson. Proofs Barrett and Feinsilver were very close to an alternative proof, but they Examples related to structured ranks only worked with tridiagonal and semiseparable matrices. References Recently also Strang and Nguyen proved a weaker formulation of the theorem. 2 Generalizations of the nullity theorem The LU -decomposition The QR -decomposition References Structured Rank Matrices Lecture 2: Structure Transport Structured Rank Matrices Lecture 2: Structure Transport 7 / 26 8 / 26 � �