Suppose S is an n ×n matrix whose columns form an...

Suppose $S$ is an $n \times n$ matrix whose columns form an orthogonal, but not orthonormal, basis of $\mathbb{R}^n$. (a) Find a formula for $S^{-1}$ mimicking the formula $Q^{-1}=Q^T$ for an orthogonal matrix. (b) Use your formula to determine the inverse of the wavelet matrix $W$ whose columns form the orthogonal wavelet basis (4.9) of $\mathbb{R}^4$.

Key Concepts

Inverse of a Matrix with Orthogonal Columns

For a square matrix S whose columns are orthogonal but not necessarily normalized, the identity S?S = D, where D is a diagonal matrix of the squared norms of the columns, holds. This relationship allows us to derive the inverse of S as S?¹ = D?¹S?. This formula generalizes the familiar result for orthogonal matrices (where D is the identity matrix) and is proven by checking that S?¹S = D?¹S?S = D?¹D = I.

Application to Wavelet Matrices

A wavelet matrix constructed from an orthogonal wavelet basis is a central object in signal processing and time-frequency analysis. When the columns of such a matrix are orthogonal but not normalized, the same reasoning applies: one must compute the squared norms of the columns to form a diagonal matrix D and then use the formula S?¹ = D?¹S? to obtain the inverse. This approach adapts the standard procedure for orthogonal matrices to account for varying lengths of the wavelet basis functions.

Orthogonal Basis

An orthogonal basis in a vector space is a set of vectors that are all perpendicular to each other. This property implies that the inner product (dot product) between any two distinct basis vectors is zero. In the context of matrices, if the columns of a matrix form an orthogonal basis, then the matrix has the property that its transpose multiplied by itself results in a diagonal matrix where each diagonal entry is the square of the norm of the corresponding column.

Gram Matrix and Diagonal Scaling

The product S?S, where S is a matrix with orthogonal columns, is called the Gram matrix and is diagonal because of the orthogonality of the columns. The diagonal entries of this matrix are the squares of the norms of the columns. This fact is crucial for deriving an inverse formula similar to the one for orthogonal (orthonormal) matrices. To adjust for the non-unit norms, one introduces a diagonal matrix D whose entries are these squared norms, leading to a modified inverse relationship.

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