Definitions

1. A unitary matrix is called special if its determinant is 1.
2. A real unitary matrix is called orthogonal.The following properties apply to orthogonal matrices as well as to unitary matrices.

Special Cases

1. Special cases include the similarity matrix P being an elementary transformation matrix, or an orthogonal matrix or a unitary matrix.

Orthonormal Basis

1. On the other hand, the representing matrix of a unitary transformation relative to an orthonormal basis is, in fact, a unitary matrix.

Rotation Matrix

1. Hence the rotation matrix is actually a unitary matrix of size 2x2 of complex entries.

Entries

1. A unitary matrix in which all entries are real is the same thing as an orthogonal matrix.
2. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries.

Orthogonal

1. The orthogonal, or QR, factorization expresses any rectangular matrix as the product of an orthogonal or unitary matrix and an upper triangular matrix.

Familiar Form

1. A more familiar form of spectral theorem is that any normal matrix can be diagonalized by a unitary matrix.

Orthogonal Matrix

1. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.

Orthonormal Bases

1. A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases.

Determinant

1. The determinant of a unitary matrix is a complex number of modulus one.

Unitary Operators

1. Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).

Matrix

1. It expresses the matrix as the product of a real orthonormal or complex unitary matrix and an upper triangular matrix.

Absolute Value

1. The determinant of a unitary matrix has an absolute value of 1.

Unitary Transformation

1. On the one hand, a unitary matrix defines a unitary transformation of relative to the inner product (2).

Unitary Matrix

1. It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e.
2. More generally, we can factor a complex m × n matrix (with m ≥ n) as the product of an m × m unitary matrix and an m × n upper triangular matrix.

Categories

1. Normal Matrix
2. Triangular Matrix
3. Unitary Transformation
4. Absolute Value
5. Unitary Operators
6. Books about "Unitary Matrix" in Amazon.com
   

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