Transpose, of a matrix

Transpose of a matrix The transpose of a matrix A, denoted by A, is formed by intereiianging its rows and eoiumns. 

The transpose of a a matrix is formed by changing each column into a row (or each row into a column). The matrix transpose of a matrix, A is denoted by the superscript T to give AT. 

The transpose of a matrix X is obtained by interchanging its rows and columns and is denoted by X. If X is an nxp matrix then X is apxn matrix. In particular, we have that ... 

T Indicates the transpose of a matrix, when used as a superscript Tt Tree i... 

In Chapters 2 and 3, we discussed the rules related to solving systems of linear equations using elementary algebraic manipulation, including simple matrix operations. The past chapters have described the inverse and transpose of a matrix in at least an introductory fashion. In this installment we would like to introduce the concepts of matrix algebra and their relationship to multiple linear regression (MLR). Let us start with the basic spectroscopic calibration relationship ... 

The transpose of a matrix or a vector is formed by assembling the elements of the first row of the matrix as the elements of the first column of the transposed matrix, the second row into the second column, and so on. In other words, atj in the original matrix A becomes the component aji in the transpose Ar. Note that the position of the diagonal components atj) are unchanged by transposition. If the dimension of A is n X m, the dimension of Ar is m X n (m rows and n columns). If square matrices A and AT are identical, A is called a symmetric matrix. The transpose of a vector x is a row... 

The transpose of a matrix is another matrix which has columns that are the same as the rows of the original matrix. 

If the transpose of a matrix is identical in every element to the original matrix (that is, if A = A), then the matrix is called a symmetric matrix. Thus, a symmetric matrix... 

Finally, some special notation for matrix operations are used throughout the text superscript t indicates the transpose of a matrix ... 

An algebraic criterion for orthogonality is stated in terms of a relation between a matrix and its transpose. The transpose of a matrix A is a matrix A such that the i, element of A is equal to the j, element of A. For example, the... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

Before defining the adjoint matrix we must define the transpose of a matrix. This is a matrix of which the columns are the rows, and vice versa, of the original matrix. Symbolically, the transpose of the matrix [ay] is [a Now, the adjoint matrix of a matrix [ay] is defined as follows ... 

If the transpose of a matrix is identical in every element to the original matrix (that is, if A = A), then the matrix is called a symmetric matrix. Thus, a symmetric matrix has all elements AtJ equal to all elements Ajt it is symmetric with respect to its principal diagonal from upper left to lower right. A symmetric matrix is necessarily a square matrix, because otherwise its transpose would have different dimensions and could not be identical to it. 

TRANSPOSE gives die transpose of a matrix. Select die destination of the correct shape, and use die syntax =TRANSPOSE(A) as illustrated in Figure A.7. 

The transpose of a matrix is obtained by exchanging rows and columns. Equation 7.1-9 for reaction 7.1-1 is... 

The transpose of a matrix, indicated by a prime ( ), is produced when rows and columns of a matrix are interchanged, i.e.. 

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