Applications of the diagonal representation III

Sep 18

Applications of the diagonal representation III

Applications of the diagonal representation III

6. Absolute value of a matrix

We know that for any matrix $A$ , the matrix $A^TA$ is symmetric and non-negative.

For a complex number $c$ the absolute value is defined by $|c|=(\bar{c}c)^{1/2}$ . 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 $A^TA$ has non-negative eigenvalues. Hence we can define the absolute value of $A$ as a square root of $A^TA,$ $|A|=(A^TA)^{1/2}.$

7. Polar form

A complex nonzero number $c$ in polar form is $c=\rho e^{i\theta}$ where $\rho >0$ is the absolute value of $c$ and $\theta$ is a real angle, so that $|e^{i\theta}|=1.$ The matrix analog of this form obtains when the condition $\rho >0$ is replaced by $\det A\neq 0,$ the absolute value of $A$ from Definition 1 is used and an orthogonal matrix plays the role of $e^{i\theta}.$

Exercise 2. For a symmetric matrix $A,$ its determinant equals the product of its eigenvalues.

Proof. $\det A=\det(Udiag[\lambda_1,...,\lambda_n]U^{-1})=(\det U)(\det diag[\lambda_1,...,\lambda_n])(\det(U^{-1}))$

$=(\det U)^2\det diag[\lambda_1,...,\lambda_n]=\lambda_1...\lambda_n.$

Exercise 3. Let $\det A\neq 0.$ Put $U=A|A|^{-1}.$ Then $U$ is orthogonal and the polar form of $A$ is $A=U|A|.$

Proof. Let $\lambda_1,...,\lambda_n$ be the eigenvalues of $A^TA.$ They are real and non-negative. In fact they are all positive because by Exercise 2 their product equals $\det(A^TA)=(\det A)^2.$

Hence, $|A|^{-1}$ exists and it is symmetric. $U$ also exists and is orthogonal: $U^TU=(|A|^{-1})^TA^TA|A|^{-1}=|A|^{-1}|A|^2|A|^{-1}=I.$ Finally, from the definition of $U,$ $A=U|A|.$