Canonical Forms : Diagonal Forms, Triangular Forms, Jordan Forms - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Linear algebra

• Linear algebra textbooks typically study vector spaces and linear transformations between them.
• Introduce bases, and one has matrix representations of the linear transformations. Under change of bases, the matrix representions change. How? Can one choose particular bases so that the corresponding matrix has some simple form?
• Linear transformations from a space into itself. Similarity: A → SAS ^-1

Matrix analysis (complex matrices, sometimes real)

• Includes all of linear algebra, and more....
• What can be said about the semigroup of square matrices, all of whose entries are positive (nonnegative)?
• What about matrices all of whose submatrices have positive determinant? (totally positive matrices)
• What about matrices all of whose principal submatrices have positive determinant? (P-matrices )
• Is there anything interesting about the matrix product A º B = [a_ij b_ij]? (the Hadamard product, a.k.a. the sophomore product)
• Matrix norms
• Functions of matrices, e.g., monotone or Hadamard matrix functions

Some fundamental facts

• (0.2.6) Matrix multiplication is not just  (inner products)
  • AB = [A col₁ (B) . . . A col_n (B)] (A acts on columns of B 
  •  (B acts on rows of A)
  • 
• (0.6.5) Any set of orthonormal vectors can be extended to an orthonormal basis.
• (0.7.2) Manipulation of block matrices, especially 2-by-2.
• (1.3.22) Eigenvalues of AB and BA
• If AB = BA (or AB = A^_T B , or AB =  oand if B has some special structure, then A probably has some special structure, too.
• A matrix A can be factored as A = BCD in many ways. Often the key to solving a matrix analysis problem is choosing a suitable factorization.

What does a matrix represent?

• A matrix A can be just a table of incoherent data, e.g., 20 rows, one for each of the constitutional regions of Italy; columns for total 2008 consumption of electric power, hectares of irrigated land, number of school children under age of 9, acres of parkland, miles of unpaved road, etc.
• A can be a table of coherent measurements that could be transformed in some way without destroying the basic information, e.g., spacecraft location or orientation data.
• A can summarize information about pairs of nodes on a graph, e.g., connection (No, Yes), directed connection (No, Which way?), capacity of a connection, etc.
• A can describe a linear algebraic object with respect to a given basis, e.g., a linear transformation, semilinear transformation, bilinear form, sesquilinear form, etc.

Many matrices can represent the same thing

• Spacecraft data: A → UAV (real orthogonal U and V : real orthogonal equivalence)
• Graph data: A → PAP^T (permutation P ; re-label nodes)
• Linear transformation: A → SAS ^-1 (similarity: change basis)
• Semilinear transformation: A → SA(consimilarity: change basis)
• Bilyinear form x ^T Ay : A → S^T AS (congruence: change basis → Sx , Sy)
• Bilinear form x^T Ay : A → U ^T AU (unitary congruence: change from one orthonormal basis to another x , y → Ux , Uy , U * U = I)
• Bilinear form x^T Ay : A → Q^T AQ (orthogonal congruence: change from one rectanormal basis to another x , y → Qx , Qy , Q^T Q = I)
• Sesquilinear form x * Ay : A → S * AS (*congruence; conjunctivity: change basis x , y → Sx , Sy)
• Sesquilinear form x * Ay : A → U * AU (unitary *congruenc: hange from one orthonormal basis to another x , y → Ux , Uy) 

Equivalence relations and subgroups

• M, N are given subgroups of GL_n (nonsingular, possibly di¤erent). Say that A ~ B if there is some M ∈ M and some N ∈ N such that B = MAN)
  • ~ is reflexive: A = IAI
  • ~ is symmetric: B = MAN) ⇒ A = M^-1BN^-1
  • ~ is transitive: If B = M₁ AN₁ and A = M₂ CN₂, then
    B = M₁ (M₂ CN₂) N₁ = (M₁ M₂) C (N₂N₁)
• Examples: GL_n, unitary group, real orthogonal group, complex orthogonal group, permutation group, group of nonsingular diagonal matrices, group of diagonal unitary matrices, group of diagonal real orthogonal matrices,....
• We might want to insist that N = f (M), but must have f (I) = I, f(M^-1) = f(M)^-1 and f(M₁M₂) = f(M₂)f(M₁)
• Examples: f (M) = M ^-1 ,   = M * (but not f (M) =  or f (M) = M^-)

The canonical form problem

• For a given equivalence relation, identify one distinguished (canonical) matrix in each equivalence class.
• Canonical matrices must be indecomposable under the equivalence relation.
• Two matrices are equivalent if and only if they are both equivalent to the same canonical matrix.
• Typically, a canonical matrix is a direct sum of indecomposable blocks with special structure.
• Sometimes the canonical form problem has a satisfactory solution (similarity or congruence), but sometimes it does not (unitary or complex orthogonal similarity).
• Rank is always an invariant: B = MAN ⇒ rank A = rank B
• Rank might be the only invariant: For M = N = GL_n (nonsingular equivalence) B = MAN if and only if rank A = rank B

The QR factorization

• (2.1.14) A ∈ M_n,m with n > m
• A = QR in which
• QR =2 M[r n,m and Q * Q = Im (orthonormal columns)
• R = [rij] ∈M_m is upper triangular and all r_ii > 0
• If rank A = m, then Q and R are uniquely determined and all r_ii > 0.
• If m = n, then Q is unitary.
• If A is real, then Q and R may be taken to be real.
• proof idea if rank A = m: Apply Gram-Schmidt process (0.6.4) to the columns of A, working from left to right.
• Equivalent version of  is unitary and   is upper triangular.
• 
• Typically used as: Qˆ * A is upper triangular 

Unitary similarity: Schur triangularization

• (2.3.1) Let λ₁ , . . . , λ_n be the eigenvalues of A ∈ M_n in any given order. There is a unitary U ∈ M_n such that A = UTU * in which is upper triangular with diagonal entries t_ii = λ_i , i = 1, . . . , n.
  
• proof: Let Ax = λ₁ x and x * x = 1. Let U = [x u₂ . . . u_n] be unitary.
• The …fIrst column of U * AU is U * Ax = λU * x = λ[x * x u*₂ x . . . u*_nx]T = λ[1 0 . . . 0]^T , so
  
• Repeat the reduction on A₁ , etc.

Schur triangularization: Some consequences

• (2.4.1) det A = λ₁ ...λ_n and tr A = λ₁ + .... + λ_n
• (2.4.3) Cayley-Hamilton Theorem: p_A(t) := det(tI - A), p_A (A) = 0
• (2.4.4) Sylvester’s Theorem on linear matrix equations: If B and C have no eigenvalues in common, then XB - CX = Y has a unique solution X for each given Y . In particular, if XB - CX = 0 then X = 0.
• (2.4.5) Uniqueness of Schur triangularization: Suppose that λ₁, . . . , λ_d are distinct and  and  are upper triangular, unitarily similar, and have the same main diagonal as Λ, then there is a block diagonal unitary matrix  conformal to Λ such that  that is,  for all i , j (unitary equivalence ).