Question

If the rows \(\mathbf{r}_{1}, \ldots, \mathbf{r}_{n}\) of the \(n \times n\) matrix \(A=\left[a_{i j}\right]\) are orthogonal, show that the \((i, j)\) -entry of \(A^{-1}\) is \(\frac{a_{j l}}{\left\|\mathbf{r}_{j}\right\|^{2}}\).

Short answer

The \((i,j)\)-entry of \( A^{-1} \) is \( \frac{a_{ji}}{\| \mathbf{r}_j \|^2} \).

Step-by-step solution

Understand Orthogonal Rows

The matrix \( A \) has orthogonal rows \( \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_n \). This means that the dot product of any two distinct rows is zero, i.e., \( \mathbf{r}_i \cdot \mathbf{r}_j = 0 \) for \( i eq j \). Additionally, the dot product of a row with itself is \( \mathbf{r}_i \cdot \mathbf{r}_i = \| \mathbf{r}_i \|^2 \).

Recognize Orthogonality Implications

A matrix with orthogonal rows implies that \( AA^T \) is diagonal. Specifically, the diagonal elements of \( AA^T \) are the squares of the norms of the rows: \( (AA^T)_{ii} = \| \mathbf{r}_i \|^2 \).

Write the Matrix Inverse Expression

Since \( A \) is orthogonal, the inverse \( A^{-1} \) is relatively straightforward to compute. For an orthogonal matrix, \( A^{-1} = D^{-1}A^T \) where \( D \) is a diagonal matrix with elements \( \| \mathbf{r}_i \|^2 \) on its diagonal.

Compute \( A^{-1} \)

The matrix \( D^{-1} \) consists of the reciprocal squares of the norms of the rows, so \( D^{-1}_{ii} = \frac{1}{\| \mathbf{r}_i \|^2} \). Now, \( A^{-1} = D^{-1}A^T \). The \((i,j)\)-entry of \( A^{-1} \) is given by \( (A^{-1})_{ij} = \frac{a_{ji}}{\| \mathbf{r}_j \|^2} \), matching the requirement that the element at \((i,j)\) is \( \frac{a_{jl}}{\| \mathbf{r}_j \|^2} \).

Consequently, confirm the inverse

The provided orthogonality condition ensures this relationship as the expression for the inverse of the matrix because the orthogonality simplifies the inverse calculation, and the correct computations were performed by transposing and scaling the matrix A.

Key concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix with rows (or columns) that are orthogonal unit vectors. Essentially, the dot product between any two distinct rows (or columns) is zero, and the dot product of a row with itself equals one. This property enforces that the matrix is both orthonormal and symmetric.
Orthogonal matrices have special properties:

• The inverse of an orthogonal matrix is its transpose: for a matrix \( A \), \( A^{-1} = A^T \).
• Orthogonal matrices preserve vector lengths. If \( \mathbf{v} \) is an n-dimensional vector, then \( \| A\mathbf{v} \| = \| \mathbf{v} \| \).

In terms of applications, orthogonal matrices are fundamental in linear transformations because they maintain the structure and shape of the transformed object while rotating it.

Matrix Inverse

The inverse of a matrix \( A \), denoted as \( A^{-1} \), is a matrix such that when multiplied by \( A \), results in the identity matrix. The main idea is: \( AA^{-1} = A^{-1}A = I \). For an orthogonal matrix, this inverse is simply its transpose, which simplifies many computations.
To find the inverse of a general matrix, it needs to be square, and it must possess a non-zero determinant. There are methods like Gaussian elimination or leveraging adjugates combined with determinants to achieve this.
For orthogonal matrices, since the rows are orthogonal and form a basis, these matrices' inverses can be evaluated efficiently. Most importantly, for diagonal matrices arising from orthogonal manipulations, finding the inverse involves simply reciprocating the diagonal elements.

Dot Product

The dot product, or scalar product, is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.
In matrix terms, if two row vectors \( \mathbf{r}_i \) and \( \mathbf{r}_j \) from matrix \( A \) are orthogonal, their dot product is zero: \( \mathbf{r}_i \cdot \mathbf{r}_j = 0 \) for \( i eq j \).

• The dot product of a vector with itself results in the square of its magnitude: \( \mathbf{r}_i \cdot \mathbf{r}_i = \| \mathbf{r}_i \|^2 \).

The dot product in an orthogonal matrix confirms the linear independence and orthogonality of rows or columns, which is crucial in constructing and understanding the structure of orthogonal matrices.

Diagonal Matrix

A diagonal matrix is a matrix where all entries outside the main diagonal are zero. These matrices are pivotal because their structure simplifies mathematical operations dramatically.
Diagonal matrices arise naturally in the context of orthogonal matrices. When we compute \( AA^T \) for a matrix \( A \) with orthogonal rows, the result is a diagonal matrix. Specifically, the entries on this diagonal are the squares of each row's norm.

• The inverse of a diagonal matrix is another diagonal matrix where each diagonal entry is simply the reciprocal of the initial entry.
• Diagonal matrices commute with other diagonal matrices, meaning the order of multiplication can be interchanged.

In linear algebra, diagonal matrices are a cornerstone for simplifying complex problems, especially within eigenvalue problems and transformations.