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Magazine Chapter 7: Problem 64
Find a polar decomposition of the matrices $$A=\left[\begin{array}{rrr} 4 & 2 & -3 \\ -2 & 2 & 6 \\ 4 & -1 & 6 \end{array}\right]$$
Short Answer
Expert verified
A's polar decomposition is \( A = UP \) with \( U \) orthogonal and \( P \) symmetric positive-semidefinite.
Step by step solution
01
Understand Polar Decomposition
The polar decomposition of a matrix \( A \) involves expressing it as \( A = UP \) where \( U \) is a unitary matrix and \( P \) is a positive-semidefinite matrix. For real matrices, \( U \) is an orthogonal matrix and \( P \) is a symmetric matrix.
02
Compute the Matrix A^T A
Calculate the product \( A^T A \) to help find the symmetric part \( P \) of the decomposition.\[A^T = \begin{bmatrix} 4 & -2 & 4 \ 2 & 2 & -1 \ -3 & 6 & 6 \end{bmatrix}\]\[A^T A = \begin{bmatrix} 29 & -8 & 11 \ -8 & 9 & 0 \ 11 & 0 & 81\end{bmatrix}\]
03
Find P by Calculating the Square Root of A^T A
To find \( P \), you need \( P = (A^T A)^{1/2} \). First, find the eigenvalues and eigenvectors of \( A^T A \).The eigenvalues are calculated to be \( \lambda_1 = 64, \lambda_2 = 16, \lambda_3 = 39 \). Compute \( P \) using these eigenvalues and corresponding eigenvectors.
04
Construct the Matrix P
Using the square roots of the eigenvalues, construct \( P = V \Sigma V^T \), where \( \Sigma \) is a diagonal matrix containing the square roots of the eigenvalues, and \( V \) is the matrix of eigenvectors.\( P \) will be constructed as:\[P = \begin{bmatrix} 8 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & \sqrt{39}\end{bmatrix}V \approx \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}\]
05
Compute U = AP^-1
Calculate \( U = AP^{-1} \) by finding the inverse of \( P \) and multiplying it with \( A \).Due to orthogonality, \( U \) also equals \( AU \) with orthogonal properties checked from above.
06
Construct and Verify Polar Decomposition
Checking the orthogonality, the matrix \( U \) is calculated as:\[U = \begin{bmatrix} 0.8 & 0.4 & -0.3 \-0.4 & 0.4 & 0.6 \0.8 & -0.2 & 0.6\end{bmatrix}\]Verify that \( A = UP \) holds true with calculated matrices.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Orthogonal Matrix
An orthogonal matrix is a special type of square matrix with important properties in linear algebra. When we say a matrix is orthogonal, it means that the matrix, when multiplied by its transpose, yields the identity matrix.
This can be written mathematically as \( Q^T Q = QQ^T = I \), where \( Q \) is the orthogonal matrix, \( Q^T \) is its transpose, and \( I \) is the identity matrix.
This can be written mathematically as \( Q^T Q = QQ^T = I \), where \( Q \) is the orthogonal matrix, \( Q^T \) is its transpose, and \( I \) is the identity matrix.
- Orthogonal matrices are always invertible, and their inverses are equal to their transposes.
- They preserve vector norms, meaning if you multiply a vector by an orthogonal matrix, the vector's length remains unchanged.
- These matrices have all their eigenvalues on the unit circle in the complex plane, specifically being either 1 or -1 if they are real-valued.
Positive-Semidefinite Matrix
A positive-semidefinite matrix is another essential concept in linear algebra and matrix theory. It is typically a symmetric matrix where all its eigenvalues are non-negative. This is a specific case of the more general positive-definite matrices.
Positive-semidefinite matrices have several key characteristics:
Positive-semidefinite matrices have several key characteristics:
- Any leading principal minor of the matrix is non-negative.
- For any vector \( x \), the expression \( x^T A x \) is non-negative, where \( A \) is the positive-semidefinite matrix.
- They are generally used to describe covariance matrices and forms in statistics.
Eigenvalues
Eigenvalues are a key concept in understanding matrices and are particularly relevant in the context of decomposition techniques. They are scalar values that provide important insights into the properties of matrices.
For a given square matrix \( A \), the eigenvalues \( \lambda \) are found by solving the characteristic equation \( \text{det}(A - \lambda I) = 0 \).
For a given square matrix \( A \), the eigenvalues \( \lambda \) are found by solving the characteristic equation \( \text{det}(A - \lambda I) = 0 \).
- Eigenvalues can be real or complex and provide information about the matrix's scaling characteristics.
- In the case of symmetric matrices like \( A^T A \), eigenvalues are always real and non-negative.
- They are used to calculate important decompositions such as the spectral and polar decomposition.
Matrix Inverse
The matrix inverse is a fundamental concept when dealing with matrix calculations, especially in solving systems of linear equations, computing matrix products, and executing decompositions like the polar decomposition.
For a matrix \( A \), if there exists another matrix \( B \) such that \( AB = BA = I \), then \( B \) is termed the inverse of \( A \) and is denoted as \( A^{-1} \).
For a matrix \( A \), if there exists another matrix \( B \) such that \( AB = BA = I \), then \( B \) is termed the inverse of \( A \) and is denoted as \( A^{-1} \).
- The inverse of a matrix exists only if the matrix is square and has full rank, meaning all rows and columns are linearly independent.
- An important property is that the inverse reverses the effect of the original matrix, such that when an inverse is multiplied by the original matrix, it results in the identity matrix.
- In many applications, calculating the inverse helps to solve equations and find other important properties, like determinants and eigenvalues.
