What is the characteristic of symmetric matrix?

What is the characteristic of symmetric matrix?

Properties of Symmetric Matrix If A and B are two symmetric matrices and they follow the commutative property, i.e. AB =BA, then the product of A and B is symmetric. If matrix A is symmetric then An is also symmetric, where n is an integer. If A is a symmetrix matrix then A-1 is also symmetric.

What is the characteristic polynomial of the identity matrix?

The characteristic polynomial of A is defined as f(X) = det(X · 1 − A), where X is the variable of the polynomial, and 1 represents the identity matrix. f(X) is a monic polynomial of degree n.

How do you define a characteristic polynomial?

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients.

What is the formula of symmetric matrix?

In linear algebra, a symmetric matrix is defined as the square matrix that is equal to its transpose matrix. The transpose matrix of any given matrix A can be given as AT. A symmetric matrix A therefore satisfies the condition, A = AT.

What is the determinant of symmetric matrix?

Symmetric Matrix Determinant Let A be the symmetric matrix, and the determinant is denoted as “det A” or |A|. Here, it refers to the determinant of the matrix A. After some linear transformations specified by the matrix, the determinant of the symmetric matrix is determined.

What is the characteristic polynomial of a square matrix?

In linear algebra, a characteristic polynomial of a square matrix is defined as a polynomial that contains the eigenvalues as roots and is invariant under matrix similarity. The characteristic equation is the equation derived by equating the characteristic polynomial to zero.

What is the characteristic polynomial of the zero matrix?

The characteristic polynomial of the zero matrix is 0.

Which of the following is a characteristic polynomial?

The characteristic polynomial of A is p(λ) = det(λI − A), whose roots are the characteristic values of A. Here, matrices are considered over the complex field to admit the possibility of complex roots. The characteristic equation, p(λ) = 0, is of degree n and has n roots.

How do you write the characteristic equation of a matrix?

Let A be any square matrix of order n x n and I be a unit matrix of same order. Then |A-λI| is called characteristic polynomial of matrix. Then the equation |A-λI| = 0 is called characteristic roots of matrix. The roots of this equation is called characteristic roots of matrix.