Question

If \(A\) is square show that \(|\operatorname{det} A|\) is the product of the singular values of \(A\).

Short answer

|det(A)| is the product of the singular values of A.

Step-by-step solution

Understand the Problem

The task is to prove that for a square matrix \(A\), the absolute value of the determinant, \(|\operatorname{det} A|\), is equal to the product of the singular values of \(A\). Singular values are derived from the singular value decomposition (SVD) of a matrix.

Recall Singular Value Decomposition

For any given square matrix \(A\), if \(A\) has a singular value decomposition, it can be expressed as \(A = U \Sigma V^*\), where \(U\) and \(V\) are unitary matrices, and \(\Sigma\) is a diagonal matrix of singular values \(\sigma_1, \sigma_2, \ldots, \sigma_n\).

Use Properties of Determinant

The determinant of a product of matrices is the product of their determinants. For a square matrix \(A\) decomposed using SVD as \(A = U \Sigma V^*\), \(\operatorname{det}(A) = \operatorname{det}(U) \cdot \operatorname{det}(\Sigma) \cdot \operatorname{det}(V^*)\).

Evaluate Determinant of Unitary Matrices

Since \(U\) and \(V^*\) are unitary matrices, their determinants are complex numbers with absolute value 1; thus, \(|\operatorname{det}(U)| = 1\) and \(|\operatorname{det}(V^*)| = 1\).

Determine the Determinant of Diagonal Matrix \( \Sigma \)

The determinant of a diagonal matrix \(\Sigma\) is the product of its diagonal entries, which are the singular values: \(\operatorname{det}(\Sigma) = \sigma_1 \sigma_2 \cdots \sigma_n\).

Conclude the Equation

Therefore, \(|\operatorname{det}(A)| = |\operatorname{det}(U) \cdot \operatorname{det}(\Sigma) \cdot \operatorname{det}(V^*)| = |1 \times (\sigma_1 \sigma_2 \cdots \sigma_n) \times 1| = \sigma_1 \sigma_2 \cdots \sigma_n\). This means \(|\operatorname{det}(A)|\) is the product of the singular values.

Key concepts

Determinant

The determinant of a matrix is a special number that can provide insight into certain properties of the matrix. For a square matrix, the determinant helps us understand if the matrix is invertible and the nature of its transformation effects.

• A matrix with a determinant of zero is singular, meaning it does not have an inverse.
• The determinant gives the scaling factor by which areas (in 2D) or volumes (in higher dimensions) change under the transformation represented by the matrix.
• If the determinant is negative, the transformation also includes a reflection.

When working with determinants in the context of singular value decomposition (SVD), we often need to consider the absolute value of the determinant, denoted \(|\operatorname{det} A|\). This is because the absolute value corresponds to the product of the singular values, ensuring all results are non-negative.
For square matrices, this connection tells us that the magnitude of the overall transformation (including any stretching, squashing, or rotating) is encoded in the singular values.

Singular Values

Singular values are key components of the singular value decomposition (SVD) which is pivotal in understanding complex matrices. SVD expresses a matrix in terms of its most important dimensions.

• Each singular value represents a distinct scaling factor for some direction in the matrix's transformation.
• SVD breaks down a matrix into three parts: two unitary matrices and one diagonal matrix of singular values.
• The main diagonal of the diagonal matrix contains the singular values, sorted in descending order. These values are always non-negative.

By linking singular values to the determinant, we can interpret them as the "volume" affected in various dimensions. This relationship is particularly useful because even if the matrix has complex behaviors, the product of these singular values will match the absolute value of the determinant, thus confirming its connection to the original matrix's characteristics like invertibility and scale.

Unitary Matrix

A unitary matrix is an essential concept in linear algebra, especially when working with complex matrices. These matrices are notable for preserving the inner product, which means they maintain the norm and angles of vectors they multiply.

• Unitary matrices have orthonormal columns, which makes them analogous to rotation matrices in real spaces.
• For any unitary matrix \(U\), it holds that \(U^* U = UU^* = I\), where \(U^*\) is the conjugate transpose and \(I\) is the identity matrix.
• Determinants of unitary matrices are special; they have an absolute value of 1. This property is crucial when decomposing a matrix into singular value decomposition (SVD).

In SVD, the presence of unitary matrices (usually denoted \(U\) and \(V^*\)) helps isolate the core scaling effects of the matrix into the diagonal matrix \(\Sigma\) containing singular values. Because their determinants have an absolute value of 1, they don't affect the absolute value of the product in our determinant calculations, allowing the singular values to encapsulate much of the matrix's transformation characteristics.