![]() Why are diagonal matrices important? Now that we have discussed the relationship between diagonal matrices and eigenvalues of operators, we can try to summarize the importance of diagonal matrices in quantum-mechanical calculations. Theorem 9 If T ∈ L ( V ) T ∈ L ( V ) has dim V d i m V distinct eigenvalues, then T T is diagonalizable. That is, can we find some basis of subspaces of Hilbert space, so the operators have matrices that contain nonzero entries only at their diagonals? To know if an operator is diagonalizable, we check it with the following conditions. In quantum-mechanical calculations, especially, people care more about finding out whether an operator on Hilbert space can be diagonalized. While the author admits that Theorem 7 is obvious logically based on our pevious dicussion, it might not be useful for solving real-world problems. v j ) is invariant under T T for each j = 1. , v n is upper triangular b) Tv j ∈∈ span ( v 1, v 2. Then the following conditions are equivalent: a) the matrix of T T w.r.t v 1. Theorem 7 Suppose T ∈ L ( V ) T ∈ L ( V ) and v 1. With this, we can write out the conditions for an operator to have upper-triangular matrix. , v j ), we should have that span ( v 1, v 2. Based on our discussion of presentation of linear map, we can deduce that for a linear map, if it has a upper-triangular matrix, its application to basis vector v j v j of vector space results in linear combination of v 1, v 2. On the other hand, if we have zero diagonal values, then the upper-triangular matrix is no longer invertible. It surely can be zero as 0 is a valid value for eigenvalue. It is noteworthy that Theorem 6 does not imply that diagonal values of the upper-triangular matrix are always nonzero. Then T T has an upper-triangular matrix with respect to some basis of V V. Theorem 6 Suppose V V is a finite-dimensional complex vector space and T ∈ L ( V ) T ∈ L ( V ). From the perspective of linear algebra, arguably the most interesting property of idempotent operation is Idempotent operators on Hilbert space is an vibrant topic in the community of quantum computing, but we will save the excitement until we hit the topic. For T ∈ L ( V ) T ∈ L ( V ), we call the operator idempotent if T 2 = T T 2 = T. Let us first discuss p ( T ) p ( T ) with a degree of two. So it's worth our time to know their interesting properties. You will find such a representation pervasive in the literature of quantum computing/algorithm. Polynomial of operator The polynomial in the proof above is a polynomial of operators. So we introduce properties of p ( T ) p ( T ) in the following section. □ The proof above uses the fundamental theorem of algebra to show the existence of eigenvalue, indicating that polynomial of operator, p ( T ) p ( T ), could be very useful. Thus, T - □ j I T - □ j I is not injective for at least one of □ j □ j, which equivalent to one of □ j □ j is eigenvalue of T T. = c ( T - □ 1 I ) ( T - □ 1 I ) ⋯ ( T - □ m I ) vĠ = a 0 v + a 1 T v + a 2 T 2 v + ⋯ + a n T n v = ( a 0 I + a 1 T + a 2 T 2 + ⋯ + a n T n ) v = c ( T - □ 1 I ) ( T - □ 1 I ) ⋯ ( T - □ m I ) v here m ≤ n m ≤ n as some complex coefficient migth be zero. = ( a 0 I + a 1 T + a 2 T 2 + ⋯ + a n T n ) v = a 0 v + a 1 Tv + a 2 T 2 v + ⋯ + a n T n v According to the fundamental theorem of algebra, we have a 0 Therefore, 0 = a 0 v + a 1 Tv + a 2 T 2 v + ⋯ + a n T n v 0 = a 0 v + a 1 T v + a 2 T 2 v + ⋯ + a n T n v has at least one of the complex coefficients a i a i being nonzero. Theorem 3 Every operator T ∈ L ( C n ) T ∈ L ( C n ) with n dimV n + 1 > d i m V. Furthermore, from Theorem 1 we have the following Suppose T ∈ L ( V ) T ∈ L ( V ) is injective, then null T = for some □ ∈ F □ ∈ F. Proof: It is obvious from a to b and c as invertibility of a map gives injectivity and surjectivity. Theorem 1 Suppose V V is finite-dimensional and T ∈ L ( V ) T ∈ L ( V ).
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |