Proving Positive Definite Scalar Product for $n \times n$ Matrices

In summary, the conversation discusses proving that the given definition of scalar product is positive definite. This involves proving two properties: (1) the scalar product of a matrix with itself is greater than or equal to 0, and (2) the scalar product is greater than 0 when the matrix is not equal to 0. The conversation mentions using the definition of trace for an $n \times n$ matrix as a starting point for the proof.
  • #1
rputra
35
0
Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product:
$$\langle X, Y \rangle = tr(X Y^T),$$
and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are
(1) $\langle X, X\rangle \geq 0$ and
(2) $\langle X, X\rangle > 0$ if $X \neq 0.$
I am lost on proving these two properties, any help or hints to prove them would be very much appreciated. Thank you before hand for your time and effort.
 
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  • #2
Hi Tarrant,

A good place to start would be with the definition of the trace for an $n\times n$ matrix. Do you recall what this was?
 

1. What is a positive definite scalar product for $n \times n$ matrices?

A positive definite scalar product for $n \times n$ matrices is a mathematical concept used to define a specific type of matrix that has important properties in linear algebra and optimization. It is a symmetric function that takes two $n \times n$ matrices as inputs and outputs a scalar value. This scalar value represents the "dot product" or "inner product" of the two matrices.

2. What are the properties of a positive definite scalar product for $n \times n$ matrices?

There are three main properties of a positive definite scalar product for $n \times n$ matrices: symmetry, positivity, and definiteness. Symmetry means that the scalar product is the same regardless of the order in which the matrices are multiplied. Positivity means that the scalar product is always greater than or equal to zero. And definiteness means that the scalar product is only equal to zero when both matrices are equal to the zero matrix.

3. How is a positive definite scalar product for $n \times n$ matrices used in linear algebra?

A positive definite scalar product for $n \times n$ matrices is used in linear algebra to define the concept of a positive definite matrix. A positive definite matrix is a specific type of matrix that has a positive definite scalar product with itself. These matrices have many important properties and are used in various applications, such as solving systems of linear equations and finding eigenvalues and eigenvectors.

4. How can one prove that a matrix is positive definite using its scalar product?

To prove that a matrix is positive definite using its scalar product, one must show that the scalar product is symmetric, positive, and definite for that particular matrix. This can be done by evaluating the scalar product for different inputs and demonstrating that it satisfies all three properties. Alternatively, one can also use the definition of a positive definite matrix and manipulate the scalar product to show that it is equivalent to the definition.

5. Are there any other important applications of a positive definite scalar product for $n \times n$ matrices?

Yes, there are many other important applications of a positive definite scalar product for $n \times n$ matrices. One such application is in optimization, particularly in quadratic optimization problems. Positive definite matrices are used to define the objective function and constraints in these problems, and the properties of the scalar product are crucial in finding optimal solutions. Additionally, positive definite matrices are also used in statistics for multivariate analysis and in machine learning algorithms.

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