## General properties of symmetric matrices

Here we consider properties of symmetric matrices that will be used to prove their diagonalizability.

### What are the differences between and

Vectors in both spaces have coordinates. In we can multiply vectors by real numbers and in - by complex numbers. This affects the notions of linear combinations, linear independence, dimension and scalar product. We indicate only the differences to watch for.

If in we multiply vectors only by real numbers, it becomes a space of dimension Let's take to see why.

**Example 1**. If we take then any complex number is a multiple of with the scaling coefficient Thus, is a one-dimensional space in this sense. On the other hand, if only multiplication by real numbers is allowed, then we can take as a basis and then and is two-dimensional. To avoid confusion, just use scaling by the right numbers.

The scalar product in is given by and in by As a result, for the second scalar product we have for complex (some people call this **antilinearity**, to distinguish it from linearity for real ).

**Definition 1**. For a matrix with possibly complex entries we denote The matrix is called an **adjoint** or a **conjugate** of

**Exercise 1**. Prove that for any

**Proof**. For complex numbers we have Therefore

Thus, when considering matrices in conjugation should be used instead of transposition. In particular, instead of symmetry the equation should be used. Matrices satisfying the last equation are called **self-adjoint**. The theory of self-adjoint matrices in is very similar to that of symmetric matrices in Keeping in mind two applications (time series analysis and optimization), we consider only square matrices with real entries. Even in this case one is forced to work with from time to time because, in general, eigenvalues can be complex numbers.

### General properties of symmetric matrices

When we extend from to is defined by the same expression as before but is allowed to be from and the scalar product in is replaced by the scalar product in The extension is denoted

**Exercise 2**. If is symmetric, then all eigenvalues of are real.

**Proof**. Suppose is an eigenvalue of Using Exercise 1 and the symmetry of we have

Since we have This shows that is real.

**Exercise 3**. If is symmetric, then it has at least one real eigenvector.

**Proof**. We know that has at least one complex eigenvalue . By Exercise 2, this eigenvalue must be real. Thus, we have with some nonzero Separating real and imaginary parts of we have with some At least one of is not zero. Thus a real eigenvector exists.

We need to generalize Exercise 3 to the case when acts in a subspace. This is done in the next two exercises.

**Definition 2**. A subspace is called an **invariant subspace** of if

**Example 2**. If is an eigenvector of then the subspace spanned by is an invariant subspace of This is because implies

**Exercise 4**. If is symmetric and is a non-trivial invariant subspace of , then has an eigenvector in

**Proof**. By the definition of an invariant subspace, the restriction of to defined by acts from to . By Exercise 3, applied to it has an eigenvector in , which is also an eigenvector of

**Exercise 5**. a) If is an (real) eigenvalue of then it is an eigenvalue of b) If is a real eigenvalue of then it is an eigenvalue of This is summarized as see the spectrum notation.

See if you can prove this yourself following the ideas used above.

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