Unitary Transformations

1954 ◽  
Vol 6 ◽  
pp. 69-72 ◽  
Author(s):  
B. E. Mitchell
Keyword(s):  
Canonical Form ◽  
Unitary Transformation ◽  
General Pattern ◽  
Jordan Form ◽  
Unitary Matrix ◽  
Complex Matrices ◽  
Triangular Form ◽  
Unitary Transformations

We consider the problem of finding a unique canonical form for complex matrices under unitary transformation, the analogue of the Jordan form (1, p. 305, §3), and of determining the transforming unitary matrix (1, p. 298, 1. 2). The term “canonical form” appears in the literature with different meanings. It might mean merely a general pattern as a triangular form (the Jacobi canonical form (8, p. 64)). Again it might mean a certain matrix which can be obtained from a given matrix only by following a specific set of instructions (1). More generally, and this is the sense in which we take it, it might mean a form that can actually be described, which is independent of the method used to obtain it, and with the property that any two matrices in this form which are unitarily equivalent are identical.

scholarly journals Regularizing algorithm for mixed matrix pencils

2017 ◽  
Vol 2 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Tetiana Klymchuk
Keyword(s):  
Canonical Form ◽  
Numerical Stability ◽  
Matrix Pencil ◽  
Mixed Matrix ◽  
Complex Matrices ◽  
Regularizing Algorithm ◽  
Unitary Transformations ◽  
Nonsingular Matrices ◽  
Matrix Pencils ◽  
Square Complex

AbstractP. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm to square complex matrices with respect to consimilarity transformations $\begin{array}{} \displaystyle A \mapsto SA{\bar S^{ - 1}} \end{array}$ and to pairs of m × n complex matrices with respect to transformations $\begin{array}{} \displaystyle (A,B) \mapsto (SAR,SB\bar R) \end{array}$, in which S and R are nonsingular matrices.

Infinite Quasi-Normal Matrices

1973 ◽  
Vol 25 (4) ◽  
pp. 820-828
Author(s):  
N. A. Wiegmann
Keyword(s):  
Canonical Form ◽  
Diagonal Matrix ◽  
Unitary Matrix ◽  
Normal Matrices ◽  
Finite Matrix ◽  
The Given

If A is a finite matrix with complex elements, and if A = AT (where AT denotes the transpose of A ), it is known (see [8] ) that there exists a unitary matrix U such that UA UT = D is a real diagonal matrix with non-negative elements which is a canonical form for A relative to the given U, UT transformation.

scholarly journals Causal and compositional structure of unitary transformations

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 511
Author(s):  
Robin Lorenz ◽  
Jonathan Barrett
Keyword(s):  
Unitary Transformation ◽  
Quantum Information Processing ◽  
Causal Structure ◽  
Quantum Circuit ◽  
New Formalism ◽  
Compositional Structure ◽  
Unitary Transformations ◽  
Output System ◽  
New Feature ◽  
Circuit Decomposition

The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system A to output system B, system A cannot influence system B. Conversely, given a unitary U with a no-influence relation from input A to output B, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of U with no path from A to B. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident simultaneously. To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A causally faithful extended circuit decomposition, representing a unitary U, is then one for which there is a path from an input A to an output B if and only if there actually is influence from A to B in U. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

scholarly journals Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Tatiana Klimchuk ◽  
Vladimir V. Sergeichuk
Keyword(s):  
Linear Algebra ◽  
Canonical Form ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Complex Matrices ◽  
Quaternion Matrices

AbstractL. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦ ˜S

scholarly journals Canonical Forms for Certain Matrices Under Unitary Congruence

1960 ◽  
Vol 12 ◽  
pp. 438-446 ◽  
Author(s):  
J W. Stander ◽  
N. A. Wiegmann
Keyword(s):  
Canonical Form ◽  
Principal Result ◽  
Diagonal Matrix ◽  
Unitary Matrix ◽  
Canonical Forms ◽  
Singular Matrix ◽  
Unitary Congruence ◽  
The Given

If A is a matrix with complex elements and if A = AT (where AT denotes the transpose of A), there exists a non-singular matrix P such that PAPT = D is a diagonal matrix (see (3), for example). It is also true (see the principal result of (5)) that for such an A there exists a unitary matrix U such that UAUT = D is a real diagonal matrix with nonnegative elements which is a canonical form for A relative to the given U, UT transformation.

The Groups of Regular Complex Polygons

1961 ◽  
Vol 13 ◽  
pp. 149-156 ◽  
Author(s):  
D. W. Crowe
Keyword(s):  
Vector Space ◽  
Unitary Transformation ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Unitary Space ◽  
Characteristic Roots ◽  
Finite Period ◽  
Set Of Points

The two-dimensional unitary space, U2, is a complex vector space of points (x, y) = (x1 + ix2, y1 + iy2), for which the distance between (x, y) and (x', y') is defined by . A unitary transformation is a linear transformation which preserves distance. A line is the set of points (x, y) satisfying some complex equation ax + by = c. A unitary transformation is a (unitary) reflection if it is of finite period n > 1 and leaves a line pointwise invariant. Thus à unitary matrix represents a reflection if its two characteristic roots are 1 and a complex nth root (n > 1) of 1.

scholarly journals Necessary and sufficient conditions for unitary similarity

1961 ◽  
Vol 2 (1) ◽  
pp. 122-126 ◽  
Author(s):  
N. A. Wiegmann
Keyword(s):  
Representation Theory ◽  
Canonical Form ◽  
Group Representation ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Necessary And Sufficient Conditions ◽  
Unitary Matrix ◽  
Unitary Similarity ◽  
Conjugate Transpose ◽  
Necessary And Sufficient

If A and B are two complex matrices and if U is a complex unitary matrix such that UAUCT = B (where UCT denotes the conjugate transpose of U), then A and B are said to be unitarily similar. Necessary and sufficient conditions that two matrices be unitarily similar have been dealt with in [5] (from the point of view of group representation theory) and in [2] (from the point of view of developing a canonical form under unitary similarity).

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