Applications of the diagonal representation III
6. Absolute value of a matrix
We know that for any matrix , the matrix is symmetric and non-negative.
For a complex number the absolute value is defined by . Since transposition of matrices is similar to conjugation of complex numbers, this leads us to the following definition.
Definition 1. By Exercise 1 from the previous post, the matrix has non-negative eigenvalues. Hence we can define the absolute value of as a square root of
7. Polar form
A complex nonzero number in polar form is where is the absolute value of and is a real angle, so that The matrix analog of this form obtains when the condition is replaced by the absolute value of from Definition 1 is used and an orthogonal matrix plays the role of
Exercise 2. For a symmetric matrix its determinant equals the product of its eigenvalues.
Exercise 3. Let Put Then is orthogonal and the polar form of is
Proof. Let be the eigenvalues of They are real and non-negative. In fact they are all positive because by Exercise 2 their product equals
Hence, exists and it is symmetric. also exists and is orthogonal: Finally, from the definition of
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