Let K>0 be a positive definite n ×n matrix. Prove that an n ×n matrix S satisfies S^T K S=I if a

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Let K>0 be a positive definite n ×n matrix. Prove that an n...

Let $K>0$ be a positive definite $n \times n$ matrix. Prove that an $n \times n$ matrix $S$ satisfies $S^T K S=\mathrm{I}$ if and only if the columns of $S$ form an orthonormal basis of $\mathbb{R}^n$ with respect to the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T K \mathbf{w}$.

Key Concepts

Matrix Representation of Inner Products

When the columns of a matrix S represent vectors in ??, the condition S^T K S = I indicates that the inner products between the columns, under the K-induced inner product, are encapsulated in the identity matrix. In this matrix form, each diagonal entry being one shows the vectors are of unit length, and each off-diagonal entry being zero shows they are orthogonal.

Equivalence of Algebraic and Geometric Conditions

The statement S^T K S = I is an algebraic characterization that is equivalent to the geometric condition of the columns of S forming an orthonormal basis with respect to the K-induced inner product. It shows how a property expressed as a matrix equation corresponds directly to the familiar geometric notion of orthonormality in an inner product space.

Orthonormal Basis

An orthonormal basis with respect to an inner product is a set of vectors that are all of unit length and mutually orthogonal under that inner product. In other words, for any pair of distinct basis vectors v and w, the inner product ?v, w? equals zero, and for any basis vector v, the norm defined by ??v, v? is one.

K-induced Inner Product

When a positive definite matrix K is used in the formula ?v, w? = v^T K w, it defines an inner product on ??. This inner product generalizes the standard dot product by 'weighting' the contributions of the components of the vectors according to the matrix K, thereby altering the geometric structure of the space in a controlled manner.

Positive Definite Matrix

A positive definite matrix is a symmetric matrix with the property that, for any nonzero vector v, the quadratic form v^T K v is positive. This ensures that the matrix induces a valid inner product when used in the form v^T K w, and it implies invertibility and well-behaved geometric properties such as defining lengths and angles in the associated inner product space.

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