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Home » Math Homework Help » Algebra Homework Help » Matric Polynomial
Matric Polynomial
An expression of the form G(x) = A0xm + A1xm-1 + …. + Am-2x2 + Am-1x + Am, where A0, A1, …. , Am-1, Am are all square matrices of the same order, is said to be a matric polynomial of degree m if A0 ≠ 0. The coefficient of the highest power of x, viz. A0 is called the leading coefficient. The matric polynomial is said to be n rowed if n is the order of each of the matric coefficients A0, A1, …. Am-1, Am G(x) is said to be a proper matric polynomial if A0 is non-singular. Also, x is called intermediate. We assume that it is commutative with every, matric coefficient.

The characteristic equation of a square matrix

Let A [aij] be an n × n square matrix. Then A – λ In, which is a matric polynomial of first degree in λ is called the characteristic matrix of A.

The determinant | A – λ In |, which is an ordinary polynomial in λ of nth degree is called the characteristic polynomial of A.

The equation | A – λ In | = 0,

is called the characteristic equation of A.

Characteristic roots or Latent roots or Eigen values

The roots of the characteristic equation | A – λ In | = 0 of A are known as the characteristic roots or latent roots or eigen values of A.

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