Unitary spaces
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1. A Hermitian form is a numerical function that satisfies certain properties relating to addition and multiplication of vectors in a complex linear space. 2. A Hermitian form is symmetric if its value is unchanged when the order of the vectors is reversed. 3. A symmetric Hermitian form is positive definite if its value is always positive, excluding the zero vector. 4. A unitary space is a complex linear space equipped with a symmetric positive definite Hermitian bilinear form, known as the scalar product, that assigns a complex number to each pair of vectors according to certain axioms.
Unitary spaces
- 1. UNITARY SPACESSchwarz inequalityPRESENTED BY: GUL I HINA ASLAM
- 2. Hermitian forms A numerical function A(x,y) is said to be Hermitian bilinear form or just a Hermitian form in x and y if it satisfies the following conditions for arbitrary x,y and z in С and arbitrary complex α. A(x+y,z) = A(x,y) + A(x,z) A(αx,y) = αA(x,y) A(x,y+z)=A(x,y) + A(x,z) A(x,αy)= ᾱA(x,y)Using induction and above conditions, we easily obtain the general formula
- 3. Hermitian-symmetricA Hermitian form A(x,y) is said to be Hermitian-symmetric (or simply symmetric) iffor arbitrary vectors x and y. Given a symmetric Hermitian form A(x, y) in an n-dimensional complex space C.We can write in A(x,y) as following with fixed basis e1 e2,. . . , en where then
- 4. i.e., the matrix of the form A(x, y) in the basis e1, . . . , en is carried into itself by transposing the matrix and replacing all its elements by their complex conjugates. Conversely, if the coefficients of a Hermitian form A(x,y) satisfy the given condition , then A(x, y) is symmetric, sinceA matrix such that will henceforth be called Hermitian-symmetric (or simply symmetric).
- 5. Positive definiteA symmetric Hermitian bilinear form A(x,y) is said to be Positive definite if A(x,y)>0 for every x≠0.Theorem. A necessary and sufficient condition for the symmetric matrix A = to define a positive definite bilinear form A(x, y) is that the descending principal minors of the matrix all be positive.
- 6. Unitary spaces“A complex linear space C is said to be a unitary space if it is equipped with a symmetric positive definite Hermitian bilinear form (x, y), called the (complex) scalar product of the vectors x and y.”i.e., if there is a rule assigning to every pair of vectors x, y ⋲C a complex number (x, y) such that a) b) c) d)Axioms a-c imply the general formula
- 7. The length of a vector. As in the real case, by the length (or norm) of a vector x in a unitary space C we mean the quantity Every nonzero vector has a positive length, and the length of the zero vector equals 0. For any complex λ, we have the equality Basic metric concepts
- 8. The Schwarz inequalityThe inequality holds for every pair of vectors x and y in C. The idea of the proof is the same as in the real case , except that we must now be careful about complex numbers. The inequality is obvious if (x, y) = 0. Thus let (x,y)≠0. clearly ,for arbitrary complex . Expanding the left-hand side, we get Let γ be the line in the complex plane determined by the origin and the complex number (x, y), and let Ś be the line symmetric to y with respect to the real axis. Suppose λ varies over the line Ś, so that λ = tz, where t is real and zis the unit vector determining the direction of Ś.
- 9. Where;Then is real, and hence so that the inequality becomesThe left-hand side of the inequality is a quadratic trinomial in t with positive coefficients, which cannot have distinct real roots, since then it would not have the same sign for all t. Therefore the discriminant of the trinomial cannot be positive, i.e., Taking the square root, we obtain
- 10. If equality holds in, then the trinomial in the left-hand side of has a unique real root (of multiplicity two). Replacing tz by λ, we find that the trinomial in the left-hand side of has the root . Therefore and hence , so that the vectors x and y differ only by a (complex)Numerical factor.
- 11. Thank you!