This paper introduces a solution to the partial eigenvalues assignment problem of high order linear systems with time delayed system in both single-input and multi-input cases using orthogonality relations between eigenvectors of the matrix polynomial. The solution requires the knowledge of only a few eigenvalues and with corresponding left eigenvectors of matrix polynomial. The numerical examples are done to illustrate the proposed method. Theorem 1: (Sarkissian, 2001) Orthogonality of the

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... nvectors of a Matrix A Let λ 1 , λ 2 ,...,λ n be the eigenvalues of a matrix A∈C n×n and let X and Ŷ be respectively the right and the left eigenvector matrices of A. Assume that {λ 1 , λ 2 ,...,λ m }∩{λ m+1 , λ m+2 ,...,λ n } = ∅ and m<n. Partition