Chapter 5: Orthogonality Flashcards
Theorem 5.3.1 Properties of inner products and norms in Rn. Let u, v, w ∈ Rn and c ∈ R. Then
(a) u · v = v · u (commutativity of inner product)
(b) u · (v + w) = u · v + u · w (distributivity of inner product)
(c) (cu) · v = c(u · v)
(d) ||v||2 = v · v
(e) kvk ≥ 0 and kvk = 0 if and only if v = 0
(f) kcvk = |c|kvk
Theorem 5.3.5 Let S = {v1, v2, . . . , vk} be an orthogonal set of non-zero vectors in Rn. Then
S is a linearly independent set.
Coefficients for orthogonal basis in
v = c1v1 + c2v2 + · · · + ckvk,
ci =(v · vi)/(vi· vi)
, for i = 1, . . . , k
Orthogonal matrix
columns for an orthonormal set
Theorem 8.2.1 A square matrix Q is orthogonal if and only if
Q^−1 = QT
Theorem 10.4.3 Let Q be an n × n matrix. The following are equivalent,
(a) Q is orthogonal
(b) ||Qx| = ||x|| for all x ∈ Rn
(c) Qx · Qy = x · y for all x, y ∈ Rn
Theorem Ex 8.2.3 Let Q be an n × n orthogonal matrix. Then,
(a) Q^−1 is orthogonal
(b) det Q = ±1
(c) If λ is an eigenvalue of Q, then |λ| = 1
(d) If Q1 and Q2 are orthogonal n × n matrices then so is Q1Q2
Definition Let A be an n × n matrix. A is orthogonally diagonalisable if
there exists an orthogonal matrix Q and a diagonal
matrix D such that Q^TAQ = D.
Theorem 8.2.4 and 5.5.7 If A is a real symmetric matrix then
a) The eigenvalues of A are all real.
b) Eigenvectors corresponding to distinct eigenvalues are orthogonal.
Theorem 8.2.2 Principal Axes Theorem If following conditions are equivalent for an n × n matrix A.
a) A has an orthonormal set of n eigenvectors
b) A is orthogonally diagonalisable
c) A is symmetric
Definition A quadratic form in n variables is a function f : Rn →
R of the form
f(x) = x
TAx = a11x^2_1 + a22x_2^2 + · · · +annx^2_n + ∑i<j 2aijxixj
where A is a symmetric n × n matrix and x∈ Rn. A is the matrix
associated with f .
x^TAx = 1 represents an ellipse if
λ1 > 0 and λ2 > 0
x^TAx = 1 has no graph if
λ1 < 0 and λ2 < 0
x^TAx = 1 represents an hyperbola if
λ1 and λ2 have opposite signs.
Positive definite square matrix
symmetric and all its eigenvalues λ are positive (negative), that is λ > 0
Negative definite square matrix
Symmetric and all its eigenvalues λ are negative, that is λ < 0
Semidefinite
If the eigenvalues λ ≥ 0 (λ ≤ 0) then the matrix positive (negative) semidefinite
Indefinite
eigenvalues take on both positive
and negative values
Positive definite iff
xTAx > 0 for every x =/= 0 in Rn
(signature is n)
Positive semidefinite iff
xTAx ≥ 0 for every x =/= 0 in Rn
(signature=rank)
Negative definite iff
x
TAx < 0 for every x 6= 0
in Rn
(signature is −n)
Rank
p + n where p is the number of positive eigenvalues and n is the number of negative eigenvalues
Signature
p-n where p is the number of positive eigenvalues and n is the number of negative eigenvalues
Negative semidefinite iff
xTAx ≤ 0 for every x =/= 0 in Rn
(signature= −rank)
Indefinite iff
xTAx takes on both positive and
negative values. (−rank <signature<rank)
Let u and v be vectors in C^n. Then u · v =
conjugate, transpose of u * v.
Theorem 8.7.3 Let A, B denote complex n × n matrices and c ∈ C Then
a) (A^H)^H = A
b) (A + B)^H = A^H + B^H
c) (cA)^H = c conjugate A^H
d) (AB)^H = B^H A^H
Hermitian matrix
A square complex matrix A is called hermitian (selfadjoint) if A^H = A.
Theorem 8.7.5 Let A be a complex n × n hermitian matrix.
a) Every eigenvalue of a hermitian matrix A is a real number.
b) The eigenvectors corresponding to distinct eigenvalues of A are orthogonal.
Unitary matrix
A square complex matrix U is called unitary if U^−1 = U^H.
Theorem 8.7.6 Let U ∈ Mn×n(C) (the space of complex n × n matrices). The following are equivalent:
a) U is unitary matrix
b) The columns of U form an orthonormal basis for Cn with respect to the complex dot product
c) ||Ux|| = ||x|| for every x ∈ Cn
d) Ux · Uy = x · y for every x, y ∈ Cn
Unitary diagonalisable
A square complex matrix A is called unitary diagonalisable if there exists a unitary matrix U and diagonal matrix D such that UH AU = D.
Spectral Theorem
Every hermitian matrix A is unitarily diagonalisable.
Are all unitarily diagonalisable matrices Hermitian?
No
Are all hermitian matrices unitarily diagonalisable?
Yes
Principal axis theorem asserts that
a real matrix is symmetric if and only if it is orthogonally diagonalisable
Are all orthogonally diagonalisable matrices symmetric?
Yes
Are all symmetric matrices orthogonally diagonalisable?
Yes
Normal matrix
A square complex matrix A is called normal if A^H A = A A^H
Theorem 8.7.9 A square complex matrix A is unitarily diagonalisable if and only if
A is normal, that is A^H A = A A^H
Matrices that are normal
Every hermitian matrix, every unitary matrix, and every skew hermitian matrix (A^H = −A) is normal.
