By Wilfred Kaplan, Donald J. Lewis
Within the moment quantity of Calculus and Linear Algebra, the concept that of linear algebra is additional constructed and utilized to geometry, many-variable calculus, and differential equations.
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Extra resources for Calculus and Linear Algebra. Volume 2: Vector Spaces, Many-Variable Calculus, and Differential Equations
Since μ = λ, we must have μ|λ = 0. Suppose λ is k-fold degenerate. Then there are k independent eigenvectors corresponding to λ. We may invoke to the Gram-Schmidt orthonormalization, for example, to obtain an orthonormal basis in this k-dimensional space. Accordingly, the set of eigenvectors of a Hermitian matrix is always chosen to be orthonormal. 3 The Pauli matrix σy = 0 −i i 0 is Hermitian. Let us ﬁnd its eigenvalues and corresponding eigenvectors. From det(σy − λI) = λ2 − 1 = 0, we ﬁnd the eigenvalues λ1 = 1 and λ2 = −1.
A good measure for this purpose is ﬁdelity, which we now deﬁne . 2 Let ρ1 and ρ2 be two density matrices belonging to the same state space S(H). 55) √ where ρ1 is chosen such that all the squre-roots of the eivenvalues are positive-semideﬁnite. A few comments are in order. • Let ρ1 = i pi |pi pi | be the spectral decomposition of ρ1 . Then the √ √ requirement in the deﬁnition claims that ρ1 = i pi |pi pi |. • F (ρ, ρ) = 1 since F (ρ, ρ) = tr √ √ ρρ ρ = tr ρ = 1. • F is non-negative by deﬁnition and F (ρ1 , ρ2 ) < 1 for ρ1 = ρ2 .
The eigenvalues of A† A are λ1 = 4 and λ2 = 0 with the corresponding eigenvectors 1 1 1 −1 |λ1 = √ , |λ2 = √ . 1 2 1 2 Unitary matrix V and the singular value matrix Σ are found from these data as ⎛ ⎞ 20 1 1 −1 and Σ = ⎝ 0 0 ⎠ . V =√ 2 1 1 00 To construct U , we need 1 1 A|λ1 = √ (1, 0, i)t 2 2 and two other vectors orthogonal to |μ1 . By inspection, we ﬁnd |μ1 = 1 |μ2 = (0, 1, 0)t and |μ3 = √ (1, 0, −i)t , 2 for example. From these vectors we construct U as ⎛ ⎞ 1 √0 1 1 U = √ ⎝0 2 0 ⎠. 2 i 0 −i 26 QUANTUM COMPUTING The reader should verify that U ΣV † really reproduces A.