scholarly journals On Differentiating a Matrix

1928 ◽  
Vol 1 (2) ◽  
pp. 111-128 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Differential Operator ◽  
Characteristic Equation ◽  
General Theorem ◽  
Matrix Differential Operator ◽  
Matrix Differential ◽  
Image Position

The theorem is well known. So also is the theorem that if concerning a determinant Λ and its reciprocal expressed by means of cofactors Aij of aij. Not quite so well known is the Cayley Hamilton theorem that a matrix X =[xij] satisfies its own characteristic equationUnlike as these three results are, they nevertheless can be looked upon as particular phases of a general theorem concerning a matrix differential operator acting upon a function of a matrix X or its transposed.

On the matrix equation (AX)n = ∥A∥I

1948 ◽  
Vol 44 (2) ◽  
pp. 292-294
Author(s):  
M. E. Grimshaw
Keyword(s):  
Characteristic Equation ◽  
Matrix Equation ◽  
Present Note ◽  
General Theorem ◽  
Circulant Matrix ◽  
Geometrical Interpretation ◽  
Diagonal Matrix ◽  
Elementary Divisors ◽  
The Matrix ◽  
Image Position

It was proved by J. Williamson (1) that when C is any circulant matrix of order n and Ω is a certain diagonal matrix thenwhere ∥ C ∥ is the determinant of C and I is the unit matrix of order n. A simpler proof was given by U. Wegner (2), and later the characteristic equation of ΩC was discussed independently by L. Toscano (3). In the present note we prove the more general theorem that there corresponds to any matrix A of order n with simple elementary divisors a solution X of the equation (AX)n = ∥ A ∥ I. The theorem has a simple geometrical interpretation; the proof is almost immediate.

scholarly journals Matrix Differentiation of the Characteristic Function

1931 ◽  
Vol 2 (4) ◽  
pp. 256-264 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Differential Operator ◽  
Characteristic Function ◽  
Matrix Differential Operator ◽  
Independent Variables ◽  
Scalar Matrix ◽  
Scalar Quantity ◽  
The Matrix ◽  
Matrix Differential

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.

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