## Applications of the diagonal representation III

### 6. Absolute value of a matrix

**Exercise 1**. For a square matrix the matrix is non-negative.

**Proof**. for any

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 and Exercise 1, 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.

**Proof**.

**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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