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Partial derivative of matrix functions with respect to a vector variable

In this paper firstly the definitions of partial derivatives of scalar functions, vector functions and matrix functions with respect to a vector variable are represented systematically. After an overview of the matrix calculus related to Kronecker products is presented. Two theorems which specify the relationship between the time derivative of a matrix and its partial derivative with respect to a vector, and the partial derivative of product of two matrices with respect to a vector, are then proved. | Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 269 – 279 Special Issue of the 30th Anniversary PARTIAL DERIVATIVE OF MATRIX FUNCTIONS WITH RESPECT TO A VECTOR VARIABLE Nguyen Van Khang Hanoi University of Technology, Vietnam Abstract. The partial derivatives of scalar functions and vector functions with respect to a vector variable are defined and used in dynamics of multibody systems. However the partial derivative of matrix functions with respect to a vector variable is also still limited. In this paper firstly the definitions of partial derivatives of scalar functions, vector functions and matrix functions with respect to a vector variable are represented systematically. After an overview of the matrix calculus related to Kronecker products is presented. Two theorems which specify the relationship between the time derivative of a matrix and its partial derivative with respect to a vector, and the partial derivative of product of two matrices with respect to a vector, are then proved. 1. INTRODUCTION The partial derivatives with respect to a vector variable of scalar functions, vector functions and matrix functions have many practical applications in dynamics and control of mechanical systems [1-10]. The partial derivatives with respect to a vector variable of scalar functions and vector functions are defined and used in dynamics of multibody systems [1- 5] also in robot dynamics [6-10]. However, the investigation of the partial derivative of matrix functions with respect to a vector variable is still limited [6]. The purpose of this paper is to review the definitions of partial derivatives of scalar function and vector function with respect to a vector variable. Based on these definitions, the concept of partial derivative of matrix function with respect to a vector variable is defined in Sec. 2. The matrix product and the Kronecker product are reviewed in Sec. 3. The proofs of two theorems which specify the relationship between the time .