On Lancaster’s Decomposition of a Matrix Differential Operator

1984 ◽  
pp. 487-490
Author(s):  
Alexander Ostrowski
Keyword(s):  
Differential Operator ◽  
Matrix Differential Operator ◽  
Matrix Differential

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 ◽  
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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.

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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