Question

If \(A\) and \(B\) are positive definite and \(r>0\) show that \(A+B\) and \(r A\) are both positive definite.

Short answer

Both \(A+B\) and \(rA\) are positive definite.

Step-by-step solution

Understanding Positive Definite Matrices

A matrix is positive definite if all its eigenvalues are positive. This can also be verified if for any non-zero vector \(x\), \(x^T A x > 0\) for matrix \(A\). We will use these principles to show that \(A+B\) and \(rA\) are positive definite.

Using Definitions for Sum of Positive Definite Matrices

Given that \(A\) and \(B\) are positive definite, for any non-zero vector \(x\), we have \(x^T A x > 0\) and \(x^T B x > 0\). To prove \(A+B\) is positive definite, consider any non-zero vector \(x\), we need to show \(x^T (A+B) x > 0\).

Proving \(A+B\) is Positive Definite

Using the property of positive definite matrices: \(x^T (A+B) x = x^T A x + x^T B x\). Since both \(x^T A x > 0\) and \(x^T B x > 0\), it follows that \(x^T (A+B) x = x^T A x + x^T B x > 0\). Thus, \(A+B\) is positive definite.

Using Scalar Multiplication on Positive Definite Matrices

Consider the scalar multiplication of a positive definite matrix \(r > 0\) and positive definite matrix \(A\). For any non-zero vector \(x\), \(x^T (rA) x = r(x^T A x)\). Given that \(x^T A x > 0\) and \(r > 0\), \(r(x^T A x) > 0\).

Proving \(rA\) is Positive Definite

As shown, \(x^T (rA) x = r(x^T A x)\) results in a positive value since both \(r > 0\) and \(x^T A x > 0\). Therefore, \(rA\) is positive definite for any positive \(r\) and positive definite matrix \(A\).

Key concepts

Matrix Eigenvalues

Understanding matrix eigenvalues is crucial when dealing with positive definite matrices. Eigenvalues are special numbers associated with a matrix, which provide insight into its properties. For a matrix to be considered positive definite, all of its eigenvalues must be positive. When this condition is met, it implies that the matrix has certain beneficial characteristics. One of these is that any non-zero vector, when multiplied with the matrix, results in a positive number.

This is mathematically represented as for any non-zero vector \(x\), we have \(x^T A x > 0\). This property is not just a mathematical curiosity; it has real-world applications in areas like optimization and stability analysis. This is because positive definiteness ensures that the matrix behaves in a "well-behaved" manner under operations like squaring or inversions. Knowing how to calculate and interpret eigenvalues enhances your ability to predict these behaviors.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and matrices. It provides the tools and framework for understanding complex systems through a structured approach. In linear algebra, matrices are used to represent linear transformations, and their properties determine how these transformations affect vector spaces.

When dealing with topics like positive definite matrices, linear algebra concepts help in understanding why certain operations, like adding matrices or performing scalar multiplication, work the way they do. Linear algebra bridges the understanding from theoretical matrix operations to practical applications. Matrices can be viewed as more than just a collection of numbers; they form the backbone of many computational tasks, from solving systems of equations to finding the shortest paths in networks. Mastery of linear algebra opens doors to vast resources in computational mathematics and beyond.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar, which is simply a real number. This operation is crucial when you consider properties like positive definiteness of matrices. Suppose a matrix \(A\) is positive definite and \(r > 0\) is a scalar. When you perform scalar multiplication, the matrix \(rA\) retains the positive definiteness because multiplying a positive number by another positive number results in a positive number.

In the context of matrices, if \(x^T A x > 0\) for any non-zero \(x\), then \(x^T (rA) x = r (x^T A x) > 0\). This shows that the operation not only scales the matrix but also scales its positive definite properties. It's a simple yet powerful tool in matrix algebra, ensuring that operations preserve necessary properties for further calculations or applications. Scalar multiplication simplifies and amplifies other operations on matrices, making it a key concept when dealing with linear transformations and matrix manipulations.