Proof of Positive Definite Matrices: Symmetric & 2x2 w/Tr & Det

In summary, the conversation discusses how to prove that a symmetric matrix with distinct positive eigenvalues is positive definite. The hint is to consider the spectral decomposition of the matrix. In the second part, the same result is proven for a 2x2 symmetric matrix with a positive trace and determinant. The hint is again to use the spectral decomposition of the matrix.
  • #1
qaz
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(i) Let A=A' be an nxn symmetric matrix with distinct eigenvalues la1, la2, ..., lan. Suppose that all eigenvalues lai > 0. Prove that A is positive definite: That is, prove that z'Az > 0 whenever z ne 0. (Hint: Consider the spectral decomposition of A.)

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(ii) Let A=A' be a 2x2 symmetric matrix with tr(A)>0 and det(A)>0. Prove that A is positive definite. (Hint: Consider the spectral decomposition of A.)

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i looked at this problem forever, nothing doing for me :cry: :confused:
 
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  • #2
anyone? please help!
 
  • #3
What is the definition of spectral decomposition (and no, I'm not asking out of ignorance)?
 

FAQ: Proof of Positive Definite Matrices: Symmetric & 2x2 w/Tr & Det

What is a positive definite matrix?

A positive definite matrix is a square matrix where all of its eigenvalues are positive. This means that it is a symmetric matrix with all positive diagonal elements and positive determinants.

How can you determine if a matrix is positive definite?

One way to determine if a matrix is positive definite is by checking if all of its eigenvalues are positive. Another way is to calculate the determinant of the matrix and see if it is positive.

What is the significance of positive definite matrices?

Positive definite matrices are important in many fields, including physics, engineering, and statistics. They have many useful properties and are often used in optimization problems and to solve systems of equations.

How can you prove that a matrix is positive definite?

To prove that a matrix is positive definite, you can use the definition of positive definite matrices, which states that all of its eigenvalues are positive. You can also use the properties of symmetric matrices, such as the fact that their eigenvalues are all real.

Can a 2x2 matrix be positive definite?

Yes, a 2x2 matrix can be positive definite. In fact, it is easier to determine if a 2x2 matrix is positive definite compared to larger matrices, as you only need to check the signs of the diagonal elements and the determinant.

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