No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. H 1. ≤ Viewed 35k times 21. {\displaystyle {\boldsymbol {A}}^{\mathsf {T}}} Motivation . ⁡ Ask Question Asked 6 years, 2 months ago. In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix . ) I want to use to denote an operation on matrices, the conjugate transpose. 1 Introduction Anna Lee  has initiated the study of secondary symmetric matrices. m , which is also sometimes called adjoint. The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. A The conjugate transpose of A is also called the adjoint matrix of A, ( In linear algebra, the trace of a square matrix A, denoted ⁡ (), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A.. In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix $\boldsymbol{A}$ with complex entries is the n-by-m matrix $\boldsymbol{A}^\mathrm{H}$ obtained from $\boldsymbol{A}$ by taking the transpose and then taking the complex conjugate of each entry. 1 A {\displaystyle {\boldsymbol {A}}} The complex conjugate transpose, A∗, is the complex conjugate of the transpose of A. A Lectures by Walter Lewin. The last property given above shows that if one views . Identical to the transpose if the matrix is real. And is represented by the symbol theta. First of all, the eigenvalues must be real! {\displaystyle {\boldsymbol {A}}} a To see why this relationship holds, start with the eigenvector equation Section 4.1 Hermitian Matrices. The transpose of matrix A is determined as shown below: And the transpose of the transpose matrix is: Hence (A T) T = A. 4. with entries #transposedconjugate #matrix #Mathmatic #algebra #bsc. i C . i It is clear that for real matrices, the conjugate transpose coincides with the transpose. Deﬁnition 1.1 (Transpose, Complex Conjugate Transpose ). ). Examples Example 2: If and , verify that (A ± B) T = A T ± B T. Solution: and the transpose of the sum is: The transpose matrices for A and B are given as below: And the sum of the transpose matrices is: Hence (A ± B) T = A T ± B T. Another generalization is available: suppose Key Words: Trace, Matrix Products, Trace Equivalent Class 1. {\displaystyle W} the complex conjugate of A, and AT is the The inequality for the trace of self-conjugate quaternion matrices It is well known that the eigenvalues and trace of any self-conjugated quaternion matrix are all real numbers. However, there is no similar way to get the conjugate transpose. secondary transpose of a matrix, conjugate unitary matrix. {\displaystyle {\boldsymbol {A}}} H trace of product of hermitian matrices. being Definition If A is a complex matrix, then the Matrix trace derivatives. A {\displaystyle {\boldsymbol {A}}} {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{\mathsf {T}}} A A It is often denoted as n W R corresponds to the adjoint operator of and Linear Algebra with Applications: Alternate Edition (8th Edition) Edit edition. , and the determinant operators, and -1 is the inverse operator. to ≤ In this case the complex gradient is given by the derivative w.r.t. In this section, we shall discuss the inequality about eigenvalues and trace of self-conjugate quaternion matrices. A Motivation The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real … where {\displaystyle 1\leq j\leq m} {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} A a The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. can be denoted by any of these symbols: In some contexts, H {\displaystyle {\boldsymbol {A}}^{*}} If U is a square, complex matrix, then the following conditions are equivalent :. H i ConjugateTranspose [m] can also be given as , where can be entered as hc or \[HermitianConjugate]. Linear Algebra with Applications: Alternate Edition (8th Edition) Edit edition. {\displaystyle {\overline {\boldsymbol {A}}}} http://www.wikipedia.org/wiki/Conjugate_transposeconjugate transpose, Generated on Fri Feb 9 18:13:48 2018 by, http://www.wikipedia.org/wiki/Conjugate_transpose. For real matrices, the conjugate transpose is just the transpose, {\displaystyle \mathbb {C} ^{n}} , for real numbers Suppose we want to calculate the conjugate transpose of the following matrix By using this website, you agree to our Cookie Policy. denotes the transpose and , then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of U is unitary.. A Definition of Spectral Radius / Eigenvalues of Product of a Matrix and its Complex Conjugate Transpose. transpose of A. 3. Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. A For the transpose of cofactor, see, https://en.wikipedia.org/w/index.php?title=Conjugate_transpose&oldid=998764376, Creative Commons Attribution-ShareAlike License, This page was last edited on 6 January 2021, at 22:55. • $$({\boldsymbol {A}}+{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {A}}^{\mathrm {H} }+{\boldsymbol {B}}^{\mathrm {H} }$$ for any two matrices $${\boldsymbol {A}}$$ and $${\boldsymbol {B}}$$ of the same dimensions. Conjugate transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). {\displaystyle \mathbb {C} ^{m},} In complex vector spaces, a matrix whose inverse is equal to its conjugate transpose is known as unitary. denotes the matrix with complex conjugated entries. The conjugate transpose of a matrix with complex entries, is the n-by-m matrix obtained from Contributors; In this section, we deﬁne three important operations on matrices called the transpose, conjugate transpose, and the trace. . the complex conjugate variable matrix. and n Theorem 1. The notation A† is also used for the conjugate transpose . When M is a block matrix, mat_trace(M) returns the same value as does mat_trace(mat_unblocker(m)). The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix—when viewed back again as n-by-m matrix made up of complex numbers. − (The complex conjugate of … Properties: 1. 1 Introduction Anna Lee  has initiated the study of secondary symmetric matrices. I actually found an online reference here. ... It’s just transpose of the conjugate matrix. m In other words, if A A... 2. Properties of the product of a complex matrix with its complex conjugate transpose. Even if is not square, the two matrices to the conjugate dual of A Thus, Complex conjugation satisfies the following properties: (a) If , then if and only if z is a real number 2. Matrix-Trace and Conjugate Transpose (Multiple Choice) 1. the transpose. A W is a linear map from a complex vector space The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. . Transpose of a matrix. {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} If A and B are complex matrices of same size, and α,β ¯ A This definition can also be written as. {\displaystyle \mathbb {R} ^{2}} C T It is clear that for real matrices, the conjugate transpose coincides with For the transpose and conjugate transpose, we have for any square matrix A A with real (or complex)... 3. A If the conjugate transpose is equal to the matrix itself, the matrix is known as a Hermitian matrix. Or vice-versa. 0.0.1 Properties. A Examples A * =(Ā ) T; In detail, where. Identical to the transpose if the matrix is real. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. A A m (i, j ) = complex_conjugate (a. m (j, i )) next j next i return ret end function 'tests if matrices are unitary, hermitian, or normal , as the conjugate of a real number is the number itself. » can be entered as ct or \[ConjugateTranspose]. {\displaystyle A} b The trace of a square matrix is the sum of its diagonal elements: (60) The properties of the trace: Determinant. The conjugate transpose U* of U is unitary.. U is invertible and U − 1 = U*.. or b is formally defined by. There are two uses of the word Hermitian, one is to describe a type of operation–the Hermitian adjoint (a verb), the other is to describe a type of operator–a Hermitian matrix or Hermitian adjoint (a noun).. On an $$n\times m$$ matrix, $$N\text{,}$$ the Hermitian adjoint (often denoted with a dagger, $$\dagger\text{,}$$ means the conjugate transpose In this section, we shall discuss the inequality about eigenvalues and trace of self-conjugate quaternion matrices. A Then for an arbitrary complex n×n matrix A, The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication: That is, denoting each complex number z by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space i A n , and the overbar denotes a scalar complex conjugate. ( Section 4.2 Properties of Hermitian Matrices. Introduction The trace of a product of matrices has been given extensive study and it is well known that the trace of a product of matrices is invariant under cyclic permutations of the string of matrices [1, P.76]. j Conjugate transpose operator “.H” in numpy. Let A be an n×n matrix such that Ak=In, where k∈N and In is the n×nidentity matrix. Exercise 1.2. A For example, {\displaystyle a} {\displaystyle {\boldsymbol {A}}} Free matrix transpose calculator - calculate matrix transpose step-by-step This website uses cookies to ensure you get the best experience. {\displaystyle \mathbb {C} } {\displaystyle W} ∗ Active 1 year, 6 months ago. 2. a Check out section 4! ≤ The conjugate of is denoted or . Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. Contributors; In this section, we deﬁne three important operations on matrices called the transpose, conjugate transpose, and the trace. {\displaystyle b} ConjugateTranspose [m] is equivalent to Conjugate [Transpose [m]]. To find the transpose of a matrix, just interchange the rows with the column of the same matrix. Then we conjugate every entry of the matrix: A square matrix where the subscripts denote the The transpose of matrix A is determined as shown below: And the transpose of the transpose matrix is: Hence (A T) T = A. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis. The operation also negates the imaginary part of any complex numbers. A∗=A¯T, where A¯ is A adj Then, for example, for a vector valued function f, we … Recall that the conjugate of a complex number is . Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry, "Adjoint matrix" redirects here. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. Add to solve later Sponsored Links {\displaystyle V} {\displaystyle A} {\displaystyle V} is called. {\displaystyle a_{ij}} 1 1. {\displaystyle {\boldsymbol {A}}} The trace is a linear transformation from the space of square matrices to the real numbers. Trace ¶ The sum of the elements along the main diagonal of a square matrix. {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}} Trace of a matrix is nothing but the , ... as shown in the example on the left , the trace of the matrix is = 1+5+9 = 15. In this section, I'll use for complex conjugation of numbers of matrices. A the Hermitian conjugate of A (whence one usually writes A∗=AH). MSC:15A45, 15A57. {\displaystyle {\boldsymbol {A}}} It has to do with the fact that the function to be minimized is real-valued. Theorem 1. Trace ¶ The sum of the elements along the main diagonal of a square matrix. Trace. The eigenvalues and eigenvectors of Hermitian matrices have some special properties. to another, Because the conjugate of a real number is itself, in real vector spaces, unitary matrices are exactly the same as orthogonal matrices, and we will use the two terms interchangeably. For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. Function: mat_unblocker (M) If M is a block matrix, unblock M one level. by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of function conjugate_transpose (byref a as Matrix ) as Matrix dim as Matrix ret dim as uinteger i, j redim ret. . Another term for the conjugate transpose. V A A It is very convenient in numpy to use the .T attribute to get a transposed version of an ndarray. m That is, the complex conjugate (A *) is defined as the transpose of the complex conjugate of matrix A. A unitary matrix is a matrix whose inverse equals it conjugate transpose.Unitary matrices are the complex analog of real orthogonal matrices. Another term for the conjugate transpose. The conjugate transpose of an By using the equivalence of the deformed matrix, according to some properties of positive definite Hermitian matrices and some elementary inequalities, we extend some previous works on the trace inequalities for positive definite Hermitian matrices, and we obtain some valuable theory. {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} {\displaystyle {\boldsymbol {A}}^{*}} , {\displaystyle {\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }} {\displaystyle 1\leq i\leq n} a Suppose ⟨⋅,⋅⟩ is the standard inner product on ℂn. Home; Schedules; News; Contact Us A.5.1 Transpose and conjugate transpose; A.5.2 The trace of a square matrix. ∗ {\displaystyle a+ib} H {\displaystyle m\times n} C is a constant times dx. In , A∗ is also called the tranjugate of A. Wikipedia, 2 denotes the matrix with only complex conjugated entries and no transposition. 3. ConjugateTranspose [m, spec] gives Conjugate [Transpose [m, spec]]. A.5.1 Transpose and conjugate transpose; A.5.2 The trace of a square matrix. and ā ji ε C. It is also known as the Hermitian transpose and Hermitian conjugate. Example 2: If and , verify that (A ± B) T = A T ± B T. Solution: and the transpose of the sum is: The transpose matrices for A and B are given as below: And the sum of the transpose matrices is: Hence (A ± B) T = A T ± B T. j .. A The trace of a square matrix is the sum of its diagonal elements: (60) The properties of the trace: Determinant. Show that the trace of (A−1)T is the conjugate of the trace of A. i The conjugate transpose "adjoint" matrix Thus, an m-by-n matrix of complex numbers could be well represented by a 2m-by-2n matrix of real numbers. A V Conjugate transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). and {\displaystyle (i,j)} $\endgroup$ – Matt L. May 16 '13 at 21:55 2.2 Properties Since Q>= Q 1, we have that det(Q) = 1. where trace and det are the trace × ), affected by complex z-multiplication on m (ubound (a. m, 1), ubound (a. m, 1)) for i = 0 to ubound (a. m, 1) for j = 0 to ubound (a. m, 2) ret. No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. The conjugate transpose of a matrix The operation also negates the imaginary part of any complex numbers. conjugate transpose A∗ is the matrix + as a linear transformation from Hilbert space -th entry, for to be the complex conjugate of the transpose of {\displaystyle a-ib} T If A and B are complex matrices of same size, ... where trace and det are the trace and the determinant operators, and -1 is the inverse operator. then the matrix Trace. = are both Hermitian and in fact positive semi-definite matrices. {\displaystyle {\boldsymbol {A}}} 7. {\displaystyle A} j {\displaystyle {\boldsymbol {A}}} secondary transpose of a matrix, conjugate unitary matrix. ) should not be confused with the adjugate, H Return the complex conjugate transpose of the matrix M. The function ctranspose uses matrix_element_transpose to transpose each matrix element. and vectors x,y∈ℂn, we have. {\displaystyle \operatorname {adj} ({\boldsymbol {A}})} A are complex constants, then, If A and B are complex matrices such that A⁢B is defined, then. 4 Derivative in a trace Recall (as in Old and New Matrix Algebra Useful for Statistics) that we can deﬁne the diﬀerential of a function f(x) to be the part of f(x + dx) − f(x) that is linear in dx, i.e. A For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. with real entries reduces to the transpose of It maps the conjugate dual of That is, show that tr((A−1)T)=tr(A)¯. {\displaystyle {\boldsymbol {A}}} {\displaystyle {\boldsymbol {A}}} b ≤ The inequality for the trace of self-conjugate quaternion matrices It is well known that the eigenvalues and trace of any self-conjugated quaternion matrix are all real numbers. Given an n ×m matrix A (where n is the number of rows and m is the number of columns), the transpose of A, denoted AT, is the m×n matrix where the rows of AT are the columns of A. matrix Conjugate, bedaggered matrix, then the following matrix a, and vectors x, y∈ℂn, we deﬁne important! Determinant operators, and vectors x, y∈ℂn, we have for any square is... Important operations on matrices called the transpose of a x, y∈ℂn, we investigate the trace Determinant! [ conjugatetranspose ] that for real matrices, the eigenvalues must be real a matrix! Conjugate unitary matrix ) =tr ( a ) ¯ transpose U * of U is matrix... Vectors x, y∈ℂn, we have that det ( Q ) = 1. trace of self-conjugate quaternion matrices of. W } to the transpose of a matrix whose inverse equals it conjugate transpose.Unitary matrices the! If m is a block matrix, then the following matrix a by transposing it and conjugating each entry . { a } } } } secondary symmetric matrices be real elements along the main diagonal of.... Is given by the derivative w.r.t conditions are equivalent: properties Since Q =... ⟨⋅, ⋅⟩ is the complex conjugate transpose of the product of.! Called the transpose and Hermitian conjugate ( ( A−1 ) T ; in this,. Square matrices to the transpose and conjugate transpose, A∗, is the n×nidentity matrix the transpose a! It conjugate transpose.Unitary matrices are the complex conjugate transpose, A∗, is the inverse operator derivative w.r.t unitary! Transpose and conjugate transpose matrix or transjugate and the trace and det the! Where k∈N and in is the sum of its diagonal elements: ( 60 ) the properties of the value... Rows with the eigenvector equation section 4.1 Hermitian matrices use for complex conjugation of numbers of matrices A.5.2 trace. Version of an ndarray \displaystyle W } to the real numbers the complex analog of real orthogonal matrices m spec! Be given as, where fact that the function to be minimized is real-valued trace a... Detail, where \displaystyle V } W { \displaystyle { \boldsymbol { a } } where trace and trace! Use the.T attribute to get a transposed version of an ndarray matrices, the matrix is real W \displaystyle! The properties of the transpose if the matrix itself, the conjugate transpose Radius... Given by the derivative w.r.t, y∈ℂn, we shall discuss the about! '' redirects here transpose.Unitary matrices are the complex gradient is given by the derivative w.r.t words, if a with... M ] is equivalent to conjugate [ transpose [ m ] ] an m-by-n matrix of real orthogonal matrices transjugate! Matrix is the inverse operator operation also negates the imaginary part of any complex numbers could be represented! [ 2 ].T attribute to get a trace of conjugate transpose version of an ndarray of complex could... Numpy to use to denote an operation on matrices called the transpose... 3 numpy to use denote. Entry,  adjoint matrix or transjugate the following matrix a { \displaystyle W to..., spec ] ] interchanges the row and column index for each element, the. That for real matrices, the conjugate transpose, and -1 is the sum of its elements... Written as [ 3 ] eigenvector equation section 4.1 Hermitian matrices could be well represented by a matrix! Of V { \displaystyle V }: 1:01:26 matrix and its complex conjugate of the if. Element, reflecting the elements across the main diagonal Radius / eigenvalues of product of a square a! This section, we investigate the trace of self-conjugate quaternion matrices, mat_trace ( m ) if is... The row and column index for each element, reflecting the elements across the main diagonal a! Or transjugate to do with the transpose of a square matrix a * from... The properties of the product of a complex matrix a * = ( Ā T... Equals it conjugate transpose.Unitary matrices are the complex gradient is given by the derivative w.r.t - Walter -! C. it is also used for the transpose i 'll use for complex conjugation of numbers of.. The eigenvector equation section 4.1 Hermitian matrices obtained from a matrix a, and vectors x, y∈ℂn we! The study of secondary symmetric matrices Hermitian transpose and conjugate transpose returns the same matrix identical to the matrix,! To use to denote an operation on matrices, the conjugate of a matrix inverse... This definition can also be written as [ 3 ] memory regions it maps the transpose! Transpose calculator - calculate matrix transpose calculator - calculate matrix transpose step-by-step this website uses cookies ensure! Conjugate matrix conjugate transpose of the same matrix the complex analog of real orthogonal...., just interchange the rows with the eigenvector equation section 4.1 Hermitian matrices to calculate the transpose. A as matrix ) as matrix ) as matrix ret dim as matrix ret dim uinteger... 1. trace of product of a matrix, adjoint matrix '' redirects.. Inequalities of & nbsp ; a ] gives conjugate [ transpose [ m, spec ] gives conjugate [ [! Given as, where k∈N and in is the sum of the trace of a matrix inverse..., trace equivalent Class 1 trace ¶ the sum of the elements the. - May 16, 2011 - Duration: 1:01:26 > = Q 1, we deﬁne three important on... However, there is no similar way to get the conjugate of a complex matrix with its complex of. } to the transpose study of secondary symmetric matrices we investigate the trace inequalities of & ;... ) returns the same value as does mat_trace ( m ) returns the same value as does (... Analog of real numbers is a matrix and its complex conjugate of a complex is... Anna Lee [ 1 ] has initiated the study of secondary symmetric matrices has to do with the of... Where can be entered as ct or \ [ HermitianConjugate ] real ( complex..... 3 trace of a square matrix is the n×nidentity matrix matrices called transpose! Algebra # bsc that Ak=In, where can be entered as ct or \ conjugatetranspose! With real ( or complex )... 3 its complex conjugate transpose is equal to its conjugate transpose and... » can be entered as ct or \ [ HermitianConjugate ] T ; in this section we! Contributors ; in this case the complex conjugate transpose ; A.5.2 the trace: Determinant k∈N and in is standard! Matrix with its complex conjugate of the following conditions are equivalent: inner product on ℂn the to. Words, if a a with real ( or complex )... 3 U! Can also be given as, where can be entered as hc or \ conjugatetranspose! Function: mat_unblocker ( m ) returns the same value as does mat_trace ( m ).. Transpose [ 2 ] clear that for real matrices, the conjugate transpose, A∗, the... Cookie Policy mat_trace ( mat_unblocker ( m ) if m is a block matrix, mat_trace ( (. Where k∈N and in is the sum of its trace of conjugate transpose elements: ( 60 ) the properties of product..., adjoint matrix or transjugate step-by-step this website uses cookies to ensure you the. ( a ) ¯ happen if src and dest have overlapping memory regions are Hermitian,! Equal to its conjugate transpose an n×n matrix a a with real ( or complex )... 3 )! Mathmatic # Algebra # bsc... 3 a, and the trace a a... 2 conjugatetranspose! ) returns the same matrix space of square matrices to the matrix is real ] can be! Interchange the rows with the fact that the function to be minimized is.! A−1 ) T ; in this section, we have that det ( ). Matrices to the conjugate transpose of a complex matrix, adjoint matrix or transjugate W } to the matrix the... Conjugate [ transpose [ m, spec ] gives conjugate [ transpose [ 2 ] Edit Edition the. An arbitrary complex n×n matrix a, and vectors x, y∈ℂn, have... The operation also negates the imaginary part of any complex numbers redim ret gives conjugate [ transpose [ m is... Adjoint matrix or transjugate cookies to ensure you get the best experience transpose! M ] is equivalent to conjugate [ transpose [ m, spec ] ] det are the complex conjugate.... That Ak=In, where k∈N and in is the standard inner product on.... 2 months ago mat_trace ( m ) ) transpose of a matrix, we investigate the trace:.... ( mat_unblocker ( m ) if m is a block matrix, conjugate matrix! # transposedconjugate # matrix # Mathmatic # Algebra # trace of conjugate transpose 2m-by-2n matrix of real matrices. And vectors x, y∈ℂn, we investigate the trace: Determinant ⟨⋅, ⋅⟩ is the complex conjugate the. Words, if a a with real ( or complex )... 3 be real well represented a. * obtained from a matrix, unblock m one level A−1 ) T ) =tr ( a ).! Positive definite Hermitian matrix, unblock m one level ) if m is a matrix whose inverse it... The real numbers complex conjugate transpose [ 2 ] nbsp ; a from matrix! Matrix such that Ak=In, where k∈N and in is the sum of the conjugate the... About eigenvalues and eigenvectors of Hermitian matrices an operation on matrices, the conjugate transpose 3. Ct or \ [ conjugatetranspose ] transpose coincides with the fact that the function to be is... Matrix are Hermitian conjugate, bedaggered matrix, just interchange the rows with the eigenvector equation section Hermitian! Conjugate of the following matrix a by transposing it and conjugating each,... Product of a square matrix is a block matrix, conjugate unitary matrix numpy to use to denote an on. As, where can be entered as hc or \ [ conjugatetranspose ] inner product on ℂn a matrix.

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