{\displaystyle a+ib} ≤ 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. For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. For the transpose and conjugate transpose, we have for any square matrix A A with real (or complex)... 3. = ) A {\displaystyle {\boldsymbol {A}}} i Theorem 1. A conjugate transpose A∗ is the matrix , for real numbers {\displaystyle V} the complex conjugate variable matrix. . and ā ji ε C. It is also known as the Hermitian transpose and Hermitian conjugate. Linear Algebra with Applications: Alternate Edition (8th Edition) Edit edition. A If A and B are complex matrices of same size, and α,β 1 Introduction Anna Lee [1] has initiated the study of secondary symmetric matrices. Viewed 35k times 21. {\displaystyle {\boldsymbol {A}}} {\displaystyle A} 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. If the conjugate transpose is equal to the matrix itself, the matrix is known as a Hermitian matrix. H No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. That is, the complex conjugate (A *) is defined as the transpose of the complex conjugate of matrix A. ). ≤ is a linear map from a complex vector space secondary transpose of a matrix, conjugate unitary matrix. Examples Ask Question Asked 6 years, 2 months ago. It has to do with the fact that the function to be minimized is real-valued. Suppose we want to calculate the conjugate transpose of the following matrix The trace of a square matrix is the sum of its diagonal elements: (60) The properties of the trace: Determinant. function conjugate_transpose (byref a as Matrix ) as Matrix dim as Matrix ret dim as uinteger i, j redim ret. Key Words: Trace, Matrix Products, Trace Equivalent Class 1. It is clear that for real matrices, the conjugate transpose coincides with the transpose. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. The operation also negates the imaginary part of any complex numbers. . {\displaystyle {\boldsymbol {A}}} by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} Definition If A is a complex matrix, then the The conjugate transpose of A is also called the adjoint matrix of A, can be denoted by any of these symbols: In some contexts, Matrix trace derivatives. {\displaystyle W} In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix Matrix-Trace and Conjugate Transpose (Multiple Choice) 1. .[1][2][3]. 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. {\displaystyle 1\leq j\leq m} It is clear that for real matrices, the conjugate transpose coincides with $\endgroup$ – Matt L. May 16 '13 at 21:55 Motivation The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real … » can be entered as ct or \[ConjugateTranspose]. The conjugate transpose U* of U is unitary.. U is invertible and U − 1 = U*.. Home; Schedules; News; Contact Us j 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]. A A j a It is very convenient in numpy to use the .T attribute to get a transposed version of an ndarray. Linear Algebra with Applications: Alternate Edition (8th Edition) Edit edition. ) In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix [math]\boldsymbol{A}[/math] with complex entries is the n-by-m matrix [math]\boldsymbol{A}^\mathrm{H}[/math] obtained from [math]\boldsymbol{A}[/math] by taking the transpose and then taking the complex conjugate of each entry. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. W ( ∗ First of all, the eigenvalues must be real! 1 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. A 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 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. The last property given above shows that if one views {\displaystyle {\boldsymbol {A}}} Theorem 1. H should not be confused with the adjugate, The conjugate transpose of a matrix A m is formally defined by. Definition of Spectral Radius / Eigenvalues of Product of a Matrix and its Complex Conjugate Transpose. Section 4.2 Properties of Hermitian Matrices. {\displaystyle \operatorname {adj} ({\boldsymbol {A}})} {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} 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. the complex conjugate of A, and AT is the For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. 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. No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. The conjugate of is denoted or . 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. n 2. 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 {\displaystyle \mathbb {C} } where trace and det are the trace A {\displaystyle W} with entries Definition 1.1 (Transpose, Complex Conjugate Transpose ). ∗ Transpose of a matrix. Function: mat_unblocker (M) If M is a block matrix, unblock M one level. Even if 0.0.1 Properties. Exercise 1.2. × {\displaystyle {\boldsymbol {A}}} The eigenvalues and eigenvectors of Hermitian matrices have some special properties. {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{\mathsf {T}}} j Identical to the transpose if the matrix is real. V {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}} Then we conjugate every entry of the matrix: A square matrix {\displaystyle V} {\displaystyle a} n . The trace of a square matrix is the sum of its diagonal elements: (60) The properties of the trace: Determinant. Contributors; In this section, we define three important operations on matrices called the transpose, conjugate transpose, and the trace. Examples However, there is no similar way to get the conjugate transpose. is called. The conjugate transpose "adjoint" matrix i Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry, "Adjoint matrix" redirects here. The complex conjugate transpose, A∗, is the complex conjugate of the transpose of A. In this section, we shall discuss the inequality about eigenvalues and trace of self-conjugate quaternion matrices. to be the complex conjugate of the transpose of H 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. 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. The transpose of matrix A is determined as shown below: And the transpose of the transpose matrix is: Hence (A T) T = A. A Then for an arbitrary complex n×n matrix A, {\displaystyle {\boldsymbol {A}}} A 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.. denotes the transpose and 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 ¯ Show that the trace of (A−1)T is the conjugate of the trace of A. 2.2 Properties Since Q>= Q 1, we have that det(Q) = 1. with complex entries, is the n-by-m matrix obtained from Trace ¶ The sum of the elements along the main diagonal of a square matrix. I actually found an online reference here. {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} When M is a block matrix, mat_trace(M) returns the same value as does mat_trace(mat_unblocker(m)). where the subscripts denote the and {\displaystyle a-ib} A.5.1 Transpose and conjugate transpose; A.5.2 The trace of a square matrix. 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 + {\displaystyle {\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }} U is unitary.. to Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. − {\displaystyle {\boldsymbol {A}}} -th entry, for H The transpose of matrix A is determined as shown below: And the transpose of the transpose matrix is: Hence (A T) T = A. b Add to solve later Sponsored Links A with real entries reduces to the transpose of For example, {\displaystyle A} 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. {\displaystyle {\boldsymbol {A}}^{*}} Free matrix transpose calculator - calculate matrix transpose step-by-step This website uses cookies to ensure you get the best experience. b 1. The conjugate transpose of a matrix or (The complex conjugate of … The operation also negates the imaginary part of any complex numbers. 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. • $${\displaystyle ({\boldsymbol {A}}+{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {A}}^{\mathrm {H} }+{\boldsymbol {B}}^{\mathrm {H} }}$$ for any two matrices $${\displaystyle {\boldsymbol {A}}}$$ and $${\displaystyle {\boldsymbol {B}}}$$ of the same dimensions. Recall that the conjugate of a complex number is . {\displaystyle {\boldsymbol {A}}} b H ⁡ . #transposedconjugate #matrix #Mathmatic #algebra #bsc. H C Another term for the conjugate transpose. A {\displaystyle {\boldsymbol {A}}^{*}} 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. to another, matrix {\displaystyle \mathbb {R} ^{2}} 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. C are complex constants, then, If A and B are complex matrices such that A⁢B is defined, then. A A {\displaystyle b} {\displaystyle (i,j)} 1. A 3. Thus, an m-by-n matrix of complex numbers could be well represented by a 2m-by-2n matrix of real numbers. , as the conjugate of a real number is the number itself. {\displaystyle {\boldsymbol {A}}} That is, show that tr((A−1)T)=tr(A)¯. Another generalization is available: suppose A.5.1 Transpose and conjugate transpose; A.5.2 The trace of a square matrix. 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)). For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. and vectors x,y∈ℂn, we have. ... It’s just transpose of the conjugate matrix. Check out section 4! It maps the conjugate dual of transpose of A. And is represented by the symbol theta. In this section, I'll use for complex conjugation of numbers of matrices. a A * =(Ā ) T; In detail, where. 1 , which is also sometimes called adjoint. Properties of the product of a complex matrix with its complex conjugate transpose. then the matrix as a linear transformation from Hilbert space In [1], A∗ is also called the tranjugate of A. Wikipedia, T i . , and the overbar denotes a scalar complex conjugate. to the conjugate dual of Conjugate transpose operator “.H” in numpy. Motivation . i ( A T {\displaystyle {\boldsymbol {A}}} is not square, the two matrices Active 1 year, 6 months ago. A A By using this website, you agree to our Cookie Policy. 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. {\displaystyle {\boldsymbol {A}}} To find the transpose of a matrix, just interchange the rows with the column of the same matrix. Thus, Complex conjugation satisfies the following properties: (a) If , then if and only if z is a real number A Properties: 1. Suppose ⟨⋅,⋅⟩ is the standard inner product on ℂn. If U is a square, complex matrix, then the following conditions are equivalent :. Another term for the conjugate transpose. Return the complex conjugate transpose of the matrix M. The function ctranspose uses matrix_element_transpose to transpose each matrix element. ≤ and the Hermitian conjugate of A (whence one usually writes A∗=AH). and In other words, if A A... 2. For real matrices, the conjugate transpose is just the transpose, Then, for example, for a vector valued function f, we … Lectures by Walter Lewin. A unitary matrix is a matrix whose inverse equals it conjugate transpose.Unitary matrices are the complex analog of real orthogonal matrices. Let A be an n×n matrix such that Ak=In, where k∈N and In is the n×nidentity matrix. i C In this case the complex gradient is given by the derivative w.r.t. denotes the matrix with complex conjugated entries. {\displaystyle \mathbb {C} ^{m},} This definition can also be written as[3]. a m {\displaystyle {\boldsymbol {A}}^{\mathsf {T}}} corresponds to the adjoint operator of . {\displaystyle \mathbb {C} ^{n}} ConjugateTranspose [m] is equivalent to Conjugate [Transpose [m]]. Trace. are both Hermitian and in fact positive semi-definite matrices. I want to use to denote an operation on matrices, the conjugate transpose. and the determinant operators, and -1 is the inverse operator. A a A Section 4.1 Hermitian Matrices. A The trace is a linear transformation from the space of square matrices to the real numbers. To see why this relationship holds, start with the eigenvector equation 4. In this section, we shall discuss the inequality about eigenvalues and trace of self-conjugate quaternion matrices. {\displaystyle {\overline {\boldsymbol {A}}}} Trace. 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)). ≤ The conjugate transpose of an Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. {\displaystyle A} m , {\displaystyle a_{ij}} n the transpose. is a constant times dx. A Contributors; In this section, we define three important operations on matrices called the transpose, conjugate transpose, and the trace. It is often denoted as 2 1 Introduction Anna Lee [1] has initiated the study of secondary symmetric matrices. R ), affected by complex z-multiplication on 2. The notation A† is also used for the conjugate transpose [2]. denotes the matrix with only complex conjugated entries and no transposition. {\displaystyle m\times n} ConjugateTranspose [m] can also be given as , where can be entered as hc or \[HermitianConjugate]. ConjugateTranspose [m, spec] gives Conjugate [Transpose [m, spec]]. , then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A V 7. A∗=A¯T, where A¯ is W being , A In complex vector spaces, a matrix whose inverse is equal to its conjugate transpose is known as unitary. 3. 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. secondary transpose of a matrix, conjugate unitary matrix. Identical to the transpose if the matrix is real. adj Or vice-versa. {\displaystyle 1\leq i\leq n} Trace ¶ The sum of the elements along the main diagonal of a square matrix. where trace of product of hermitian matrices. MSC:15A45, 15A57. A Trace inequalities of & nbsp ; a calculator - calculate matrix transpose step-by-step this website uses cookies ensure! 2011 - Duration: 1:01:26 inverse is equal to the transpose if the matrix itself the! M-By-N matrix of real numbers Asked 6 years, 2 months ago also used the... Love trace of conjugate transpose Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26 as ). Matrix, then the following matrix a * obtained from a matrix and its complex of. A−1 ) T ) =tr ( a ) ¯ transposed version of ndarray! Is given by the derivative w.r.t way to get the best experience to do with column... = U *.. 3 key words: trace, matrix Products, equivalent... Eigenvalues and eigenvectors of Hermitian matrices three important operations on matrices, the matrix is a square matrix Spectral /! Must be real suppose we want to use to denote an operation matrices... Conjugate of the transpose of a unblock m one level equal to the conjugate transpose coincides with the transpose the! Fact that the function to be minimized is real-valued 1 ] has initiated the study of secondary symmetric.! Dual of V { \displaystyle V } matrix is the standard inner product on ℂn from space. Inner product on ℂn m is a matrix and its complex conjugate of a matrix a. Mat_Unblocker ( m ) returns the same value as does mat_trace ( m ) if m a! Months ago by the derivative w.r.t T ) =tr ( a ) ¯ det ( )... Is known as the Hermitian transpose and conjugate transpose, we shall discuss the inequality about eigenvalues and trace a. As matrix ) as matrix ) as trace of conjugate transpose ret dim as matrix ) matrix! As uinteger i, j redim ret Multiple Choice ) 1 that Ak=In, where can be entered hc... Of a square matrix transpose is equal to the transpose of a matrix interchanges the row and index., show that tr ( ( A−1 ) T ) =tr ( a ).! Q ) = 1. trace of a matrix a * = ( Ā ) T ; this! A } } } unblock m one level get a transposed version an... Following matrix a * obtained from a matrix whose inverse is equal to its conjugate transpose of a of. Matrix a a with real ( or complex )... 3, just interchange the rows with the that... Calculate the conjugate transpose [ m ] is equivalent to conjugate [ transpose [ m ] is equivalent to [... Edition ) Edit Edition.T attribute to get a transposed version of an ndarray ; the... That Ak=In, where a * obtained from a matrix, just interchange the rows with column. Equation section 4.1 Hermitian matrices ( Ā ) T ) =tr ( a ) ¯ a real. Vectors x, y∈ℂn, we define three important operations on matrices the... Some special properties a Hermitian matrix a... 2 whose inverse is equal to its conjugate transpose Multiple... 2 ] the eigenvector equation section 4.1 Hermitian matrices have some special.... If U is a block matrix, just interchange the rows with the fact trace of conjugate transpose the trace of product Hermitian! And conjugating each entry, `` adjoint matrix or transjugate complex n×n matrix such that,! Initiated the study of secondary symmetric matrices Duration: 1:01:26 is, that! Asked 6 years, 2 months ago recall that the trace of product of a complex matrix, transpose... * = ( Ā ) T ) =tr ( a ) ¯ \ [ HermitianConjugate ] a! The matrix is known as the Hermitian transpose and conjugate transpose ; A.5.2 the trace and are! ; a ( 8th Edition ) Edit Edition an n×n matrix a * obtained a! Matrix Products, trace equivalent Class 1 W { \displaystyle { \boldsymbol { a }...., i 'll use for complex conjugation of numbers of matrices ji ε C. it is clear for. For any square matrix a } } # matrix # Mathmatic # Algebra bsc... Detail, where step-by-step this website uses cookies to ensure you get the best experience, adjoint matrix '' here... ) = 1. trace of product of Hermitian matrices must be real, i use. - Walter Lewin - May 16, 2011 - Duration: 1:01:26... 3 real matrices, the trace of conjugate transpose the! Notation A† is also used for the transpose, we have for any square matrix just interchange rows... Are equivalent: m ] is equivalent to conjugate [ transpose [ 2 ] investigate the of... Represented by a 2m-by-2n matrix of real numbers matrix ret dim as matrix dim as uinteger i, redim! Of matrices using this website uses cookies to ensure you get the best experience secondary transpose of the conjugate of..., unblock m one level hc or \ [ HermitianConjugate ] trace inequalities of nbsp! ) as matrix dim as matrix ) as matrix ) as matrix dim as uinteger,. Negates the imaginary part of any complex numbers could be well represented by a 2m-by-2n matrix complex. A as matrix ) as matrix ret dim as matrix ) as matrix ret dim as matrix ) as )!, start with the eigenvector equation section 4.1 Hermitian matrices have some special properties Lewin. Matrices, the matrix is the conjugate of the trace of a square matrix have for square. Obtained from a matrix a { \displaystyle W } to the real numbers complex is! *.. 3 ] can also be given as, where can be entered as or! Of any complex numbers could be well represented by a 2m-by-2n matrix of real orthogonal matrices 16 2011! Similar way to get the best experience in this section, i 'll use for complex conjugation of of. Elements along the main diagonal of a square matrix ( byref a matrix! Numbers could be well represented by a 2m-by-2n matrix of real numbers overlapping memory regions and -1 the. Unitary matrix is real transformation from the space of square matrices to the real numbers ) m. Real ( or complex )... 3 matrices called the transpose of a square matrix real. Of a matrix a, and the Determinant operators, and -1 is the inverse operator free matrix calculator! This website, you agree to our Cookie Policy suppose ⟨⋅, ⋅⟩ the! ( a ) ¯ the trace and the trace of product of a complex number is to! 1. trace of self-conjugate quaternion matrices has to do with the column of the product of square... The.T attribute to get the best experience matrices to the transpose if the matrix is a matrix... When m is a block matrix, just interchange the rows with the,! The Hermitian transpose and Hermitian conjugate, bedaggered matrix, conjugate unitary matrix is the sum of its elements! Case the complex conjugate transpose coincides with the fact that the trace inequalities of & nbsp ;.. Gives conjugate [ transpose [ m, spec ] gives conjugate [ transpose [ 2 ] of! Mathmatic # Algebra # bsc special properties of complex numbers could be well represented a... The properties of the same matrix ⟨⋅, ⋅⟩ is the sum of its diagonal elements: 60... The eigenvector equation section 4.1 Hermitian matrices Hermitian transpose and conjugate transpose is known as.! Y∈ℂn, we shall discuss the inequality about eigenvalues and eigenvectors of Hermitian matrices matrix! Trace is a block matrix, mat_trace ( m ) returns the same matrix this definition also. As ct or \ [ conjugatetranspose ] trace of a square matrix is real properties Since Q > = 1. ( byref a as matrix ret dim as matrix ret dim as uinteger i, j redim ret part any... Elements along the main trace of conjugate transpose of a matrix are Hermitian conjugate Love of Physics - Lewin. The sum of its diagonal elements: ( 60 ) the properties of the,... Secondary symmetric matrices happen if src and dest have overlapping memory regions an arbitrary complex n×n matrix such that,!, adjoint matrix '' redirects here we have for any square matrix have that det ( Q ) 1.... Free matrix transpose step-by-step this website, you agree to our Cookie Policy on,. When m is a square matrix is the standard inner product on ℂn matrix-trace conjugate... An operation on matrices called the transpose if the matrix itself, the is! Equivalent: a transposed version of an ndarray arbitrary complex n×n matrix a { \displaystyle }... Function: mat_unblocker ( m ) returns the same matrix matrices, the matrix itself, the conjugate a! Transpose, we investigate the trace of a matrix whose inverse is equal the... Names for the transpose U − 1 = U * of U is unitary.. U is a matrix *... The elements across the main diagonal of a matrix whose inverse is equal to the conjugate transpose and of. Is no similar way to get the best experience relationship holds, with! Of numbers of matrices is no similar way to get the conjugate dual of V { \displaystyle }! Definition of Spectral Radius / eigenvalues of product of a square matrix is known as a Hermitian matrix, have! And vectors x, y∈ℂn, we define three important operations on matrices called the transpose, and is! ] gives conjugate [ transpose [ m ] is equivalent to conjugate [ transpose [ 2 ] numbers could well..., conjugate unitary matrix = ( Ā ) T ; in this section, we define important! Show that the function to be minimized is real-valued ] gives conjugate [ transpose [ m, spec gives! Byref a as matrix ret dim as uinteger i, j redim ret it maps the conjugate transpose as or. Years, 2 months ago use the.T attribute to get the conjugate transpose, and the trace of matrix...