Publish On: 2019-03-15

# Andy More

Total Post: 752

## Question: Describe the real quadratic form of a matrix

Reply On: 2013-12-02

# Mjay Jollay

Total Post: 0

## ANS: Describe the real quadratic form of a matrix

An expression of the form Σ Σa

Following are some facts about a real quadratic form.

A real quadratic form:

(i) is usually denoted by Q(x

(ii) has rank r if its matrix has rank r

(iii) has signature m if its matrix has signature m

(iv) is positive definite if Q(x

(v) is negative definite if Q(x

(vi) is positive semidefinite if Q(x

(vii) is negative semidefinite if Q(x

(viii) is indefinite if Q(x

_{ij}x_{i}x_{j}, where a_{ij}ϵ ℝ and a_{ij}= a_{ji}, is called a real quadratic form in the variables x_{1}, x_{2}, …., x_{n}. A real quadratic form can be written as X^{t}AX, where X = (x_{1}, x_{2}, …., x_{n})^{t}and A is a symmetric matrix, (called the matrix of the quadratic form) asFollowing are some facts about a real quadratic form.

A real quadratic form:

(i) is usually denoted by Q(x

_{1}, x_{2}, …, x_{n}), where x_{i }are the variables(ii) has rank r if its matrix has rank r

(iii) has signature m if its matrix has signature m

(iv) is positive definite if Q(x

_{1}, x_{2}, …., x_{n}) > 0 for all values of x_{1}, x_{2}, ….., x_{n}(v) is negative definite if Q(x

_{1}, x_{2}, …., x_{n}) < 0 for all values of x_{1}, x_{2}, ….., x_{n}(vi) is positive semidefinite if Q(x

_{1}, x_{2}, …., x_{n}) > 0 for some values of x_{1}, x_{2}, …., x_{n}and zero for other values of x_{1}, x_{2}, …., x_{n}(vii) is negative semidefinite if Q(x

_{1}, x_{2}, …., x_{n}) < 0 for some values of x_{1}, x_{2}, …., x_{n}and zero for other values of x_{1}, x_{2}, …., x_{n}.(viii) is indefinite if Q(x

_{1}, x_{2}, …., x_{n}) ≥ 0 for some values of x_{1}, x_{2}, …., x_{n}and Q(x_{1}, x_{2}, …., x_{n}) < 0 for other values of x_{1}, x_{2}, …., x_{n}.Like Us On Facebook for All Latest Updates