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Questions and Answers
The matrix of a quadratic form is a symmetric matrix.
The matrix of a quadratic form is a symmetric matrix.
True (A)
A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix.
A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix.
True (A)
The principal axes of a quadratic form $X^TAx$ are eigenvectors of A.
The principal axes of a quadratic form $X^TAx$ are eigenvectors of A.
True (A)
A positive definite quadratic form Q satisfies $Q(x) > 0$ for all x in $R^n$.
A positive definite quadratic form Q satisfies $Q(x) > 0$ for all x in $R^n$.
If the eigenvalues of a symmetric matrix A are all positive, then the quadratic form $x^TAx$ is positive definite.
If the eigenvalues of a symmetric matrix A are all positive, then the quadratic form $x^TAx$ is positive definite.
The expression absolute value of $x^2$ is not a quadratic form.
The expression absolute value of $x^2$ is not a quadratic form.
If A is symmetric and P is an orthogonal matrix, then the change of variable $x = Py$ transforms $x^TAx$ into a quadratic form with no cross-product term.
If A is symmetric and P is an orthogonal matrix, then the change of variable $x = Py$ transforms $x^TAx$ into a quadratic form with no cross-product term.
An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.
An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.
If A is symmetric and the quadratic form $x^TAx$ has only negative values for $x
eq 0$, then the eigenvalues of A are positive.
If A is symmetric and the quadratic form $x^TAx$ has only negative values for $x eq 0$, then the eigenvalues of A are positive.
What is the quadratic form $Q(x)$?
What is the quadratic form $Q(x)$?
Define positive definite.
Define positive definite.
Define negative definite.
Define negative definite.
Define indefinite.
Define indefinite.
Define positive semi-definite.
Define positive semi-definite.
Define negative semi-definite.
Define negative semi-definite.
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Study Notes
Quadratic Forms and Matrices
- Quadratic forms are represented as ( Q(x) = x^TAx ) where ( A ) is a symmetric matrix.
- A matrix of a quadratic form is always symmetric.
- A diagonal matrix indicates a quadratic form has no cross-product terms, indicating independence in the variables.
Properties of Quadratic Forms
- The principal axes of a quadratic form correspond to the eigenvectors of the matrix ( A ).
- A positive definite quadratic form, while it can have values greater than zero, equals zero when the vector ( x ) is the zero vector.
- Eigenvalues are crucial in determining the type of quadratic form:
- If all eigenvalues of a symmetric matrix are positive, the quadratic form is positive definite.
- A quadratic form is negative definite if it produces only negative values for any non-zero vector ( x ).
Types of Definite and Indefinite Forms
- Positive definite forms require that ( Q(x) > 0 ) for all vectors ( x \neq 0 ).
- Negative definite forms yield ( Q(x) < 0 ) for all vectors ( x \neq 0 ).
- Indefinite forms possess both positive and negative eigenvalues, confirming their mixed nature.
Semi-Definite Forms
- A quadratic form is positive semi-definite if ( Q(x) \geq 0 ) for all ( x ).
- Negative semi-definite forms satisfy ( Q(x) \leq 0 ).
Misconceptions and Exceptions
- The expression ( |x|^2 ) is not classified as a quadratic form due to its structure.
- Changing variables with an orthogonal matrix does not always eliminate cross-product terms; certain configurations can yield degenerate cases with reduced dimensionality.
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