In linear algebra, a unitary matrix is a complex square matrix whose row and column vectors are orthonormal with respect to the standard scalar product. Thus the inverse of a unitary matrix is also its adjoint .
By multiplying by a unitary matrix, both the Euclidean norm and the standard scalar product of two vectors are preserved. Every unitary mapping between two finite-dimensional scalar product spaces can be represented by a unitary matrix, depending on the choice of an orthonormal basis . The set of unitary matrices of a fixed size forms the unitary group with the matrix multiplication as a link .
Unitary matrices are used in singular value decomposition , discrete Fourier transformation and in quantum mechanics , among other things . A real unitary matrix is called an orthogonal matrix .
definition
A complex square matrix is called unitary if the product with its adjoint matrix results in the unit matrix , i.e.
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![{\ displaystyle U ^ {H} U = I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe85132b2d2a0a9e26c5557629963b3d450533)
applies. Are the column vectors of the matrix with designated, this condition is equivalent to saying that there is always the standard scalar of two column vectors
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![u_1, \ ldots, u_n](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d1c3b35be67793eec544d31fdee27419591b23)
![u_ {i} ^ {H} \ cdot u_ {j} = \ delta _ {{ij}} = {\ begin {cases} 1 & {\ text {if}} ~ i = j \\ 0 & {\ text {otherwise }} \ end {cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17542390378c24f9f0219551879c3690aab869d6)
results, where the Kronecker delta is. The column vectors of a unitary matrix thus form an orthonormal basis of the coordinate space . This also applies to the row vectors of a unitary matrix, because with the transposed matrix is also unitary.
![{\ displaystyle \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![U ^ T](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3e0e44a9cce4f8cd3dee9534ee3e6a60c1855a)
Examples
The matrix
![U = \ begin {pmatrix} 0 & i \\ i & 0 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe0583c5034a1d9c0657ce8a91d5f6201a0ab93)
is unitary because it applies
-
.
The matrix too
![U = \ frac {1} {2} \ begin {pmatrix} 1 + i & 1-i \\ 1-i & 1 + i \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68626099ae7c1c9bff11f3e8c2e2d8b2358926ac)
is unitary because it applies
-
.
In general, every orthogonal matrix is unitary, because for matrices with real entries the adjoint corresponds to the transpose.
properties
Inverse
A unitary matrix is always regular due to the linear independence of its row and column vectors . The inverse of a unitary matrix is equal to its adjoint, that is, it applies
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
-
.
The inverse of a matrix is precisely the matrix for which
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![U ^ {- 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eff2182e163ff05e8d17100b65a7ddc9a25755d7)
![U \, U ^ {- 1} = U ^ {- 1} \, U = I](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dffa61ad460e38191ec4c097132591b97c5bba7)
applies. The converse is also true, and every matrix whose adjoint is equal to its inverse is unitary, because then it is true
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
-
.
In addition, the adjoint of a unitary matrix is also unitary, because
-
.
Invariance of norm and scalar product
If a vector is multiplied by a unitary matrix , the Euclidean norm of the vector does not change, that is
![{\ displaystyle x \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d487cadb233e86b6a5c373836a6fe5a8d22c55c)
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
-
.
Furthermore, the standard scalar product of two vectors is invariant with respect to the multiplication with a unitary matrix , i.e.
![{\ displaystyle x, y \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e04de19308fcc5e285b85218e902db0c59face0)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
-
.
Both properties follow directly from the displacement property of the standard scalar product. Hence the figure represents
![{\ displaystyle f \ colon \ mathbb {C} ^ {n} \ to \ mathbb {C} ^ {n}, \ quad x \ mapsto U \, x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f108216d3c23c4cf85fe8fd63318998b9e8c9a)
represents a congruence map in unitary space . Conversely, the mapping matrix is unitary with respect to the standard basis of any linear map im that receives the standard scalar product. Because of the polarization formula , this also applies to the mapping matrix of every linear mapping that receives the Euclidean norm.
![{\ displaystyle \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![{\ displaystyle \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
Determinant
For the complex amount of the determinant of a unitary matrix we have
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
-
,
which with the help of the determinants product set over
![\ det U \ cdot \ overline {\ det U} = \ det U \ cdot \ det \ bar {U} = \ det U \ cdot \ det U ^ H = \ det (UU ^ H) = \ det I = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb39a27941cc0322839c8ef92dad5cea4b1ac016)
follows.
Eigenvalues
The eigenvalues of a unitary matrix also all have the absolute value one, so they are of the form
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
![\ lambda = e ^ {it}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7713c51c38e79510db6008e4d15b2a8b4c023937)
with . If there is an eigenvector that belongs to it, then due to the invariance with regard to the Euclidean norm and the absolute homogeneity of a norm , the following applies![t \ in \ R](https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![\ | x \ | _2 = \ | U \, x \ | _2 = \ | \ lambda \, x \ | _2 = | \ lambda | \, \ | x \ | _2](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9747e4de120fbe3cd1b7e64a9fe1d887250384)
and therefore .
![| \ lambda | = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f99c7b1dc3696f6e900ea51d946e4da3915f7e)
Diagonalisability
A unitary matrix is normal , that is, it holds
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
-
,
and therefore diagonalizable . After the spectral theorem, there is another unitary matrix such that
![{\ displaystyle V \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f94708aaf41b22c485bd842405104ab18be3426e)
![V ^ {- 1} \, U \, V = D](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0730145356e5cf2f27c33e988b1909b15e63ed)
holds, where is a diagonal matrix with the eigenvalues of . The column vectors of are then pairwise orthonormal eigenvectors of . Thus the eigenspaces of a unitary matrix are also orthogonal in pairs.
![{\ displaystyle D \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f56f65a7a69c20de0b6457c8dc840825257bb42)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
Norms
The spectral norm of a unitary matrix is
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
-
.
For the Frobenius norm , the Frobenius scalar product applies accordingly
-
.
The product with a unitary matrix receives both the spectral norm and the Frobenius norm of a given matrix , because it holds
![{\ displaystyle A \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b3023de9ab6f59e511d7c8ae72d03b64bcecbc)
![\ | U \, A \ | _2 = \ max _ {\ | x \ | _2 = 1} \ | U \, A \, x \ | _2 = \ max _ {\ | x \ | _2 = 1} \ | A \, x \ | _2 = \ | A \ | _2](https://wikimedia.org/api/rest_v1/media/math/render/svg/63a9e5ec58a48311d982998c254b36e03a178eaf)
and
-
.
This means that the condition of a matrix with regard to these norms is retained after multiplication with a unitary matrix.
Preservation of idempotence
If there is a unitary and an idempotent matrix , therefore , then the matrix is
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
![{\ displaystyle A \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b3023de9ab6f59e511d7c8ae72d03b64bcecbc)
![A \, A = A](https://wikimedia.org/api/rest_v1/media/math/render/svg/468d1612d2e2a5f6a0edede3845daa99982f7cfc)
![B = U \, A \, U ^ H](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce8bb26bdae0718fa25ea2bf434cd842ed1a88e)
also idempotent, because
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.
Unitary matrices as a group
The set of regular matrices of fixed size and the matrix multiplication as a link form a group , the general linear group . The identity matrix serves as a neutral element . The unitary matrices form a subgroup of the general linear group, the unitary group . The product of two unitary matrices is unitary again because it holds
![{\ displaystyle \ mathrm {GL} (n, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37173423b5c9e3a7b577c77869ba9b30af66ecc8)
![\ mathrm U (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a32fa84df5de5dfa91b6bdc88fb03fc8792c9f81)
![{\ displaystyle U, V \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2974b481bd0fa53045e16cf9d589f64bce7da24d)
-
.
Furthermore, the inverse of a unitary matrix is also unitary because it holds
![{\ displaystyle U \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5456850376e308334cb09a4ea84e299b3e5bc21d)
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.
The unitary matrices with determinant one in turn form a subgroup of the unitary group, the special unitary group . The unitary matrices with determinant minus one do not form a subgroup of the unitary group, because they lack the neutral element, only a secondary class .
![\ mathrm {SU} (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a205091aabd5690efdfeb7354a55844f2eb31b)
use
Matrix decompositions
With the help of a singular value decomposition , each matrix can be converted into a product
![{\ displaystyle A \ in \ mathbb {C} ^ {m \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca24dc9d0669dc657b9e98d1cd72204aa7416ac)
![A = U \, \ Sigma \, V ^ H](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d189836d9ab46223e118d71924436b2fcdeb14f)
a unitary matrix , a diagonal matrix and the adjoints of another unitary matrix . The diagonal entries of the matrix are then the singular values of .
![{\ displaystyle \ Sigma \ in \ mathbb {C} ^ {m \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0861dcf80f5cd1086f69e757906eba42f4f873)
![{\ displaystyle V \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f94708aaf41b22c485bd842405104ab18be3426e)
![\ Sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
A square matrix can also be used as a product
by means of the polar decomposition![{\ displaystyle A \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b3023de9ab6f59e511d7c8ae72d03b64bcecbc)
![A = U \, P](https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e27bde5076688517508249c2408aec9f4988c)
a unitary matrix and a positive semidefinite Hermitian matrix can be factored.
![{\ displaystyle P \ in \ mathbb {C} ^ {n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a21849762c9f9e42d9f1c3c868dd39910ad1a2e)
Unitary mappings
If a -dimensional complex scalar product space , then every linear mapping after choosing an orthonormal basis for can be determined by the mapping matrix![(V, \ langle \ cdot, \ cdot \ rangle)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07d89cdf91d100d247a9b5e39f2b4249ad7134e)
![f \ colon V \ to V](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdad235875fd740596e229f36847d469955d96c)
![\ {e_ {1}, \ dotsc, e_ {n} \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4c237dc5200c8760593d5c01530e6410a25e86)
![A_ {f} = (a _ {{ij}}) \ in \ mathbb {R} ^ {{n \ times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc4b52dce22f97cdf9b7db1ff89b5f2a7cda99b)
represent, where for is. The mapping matrix is now unitary if and only if is a unitary mapping . This follows from
![{\ displaystyle f (e_ {j}) = a_ {1j} e_ {1} + \ dotsb + a_ {nj} e_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/009649893adb930e7ba586389768422ad89ff37f)
![{\ displaystyle j = 1, \ dotsc, n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2ddca4fb9adff031ab816398e5549ff396dbf3)
![A_ {f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9cba7c44dc1773496e4ffc9c7e0be19c591369)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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,
where and are.
![{\ displaystyle v = x_ {1} e_ {1} + \ dotsb + x_ {n} e_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/871d0b120728a3bc16a4966101e12969e2a2c87c)
![{\ displaystyle w = y_ {1} e_ {1} + \ dotsb + y_ {n} e_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07df66c57da27f92e7bcac59f554aff7699f173a)
Physical applications
Unitary matrices are also often used in quantum mechanics within the framework of matrix mechanics . Examples are:
Another important application of unitary matrices is the discrete Fourier transform of complex signals.
literature
Web links