antilinearity - Raising the bar

Sep 18

General properties of symmetric matrices

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 $R^n$ and $C^n?$

Vectors in both spaces have $n$ coordinates. In $R^n$ we can multiply vectors by real numbers and in $C^n$ - 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 $C^n$ we multiply vectors only by real numbers, it becomes a space of dimension $2n.$ Let's take $n=1$ to see why.

Example 1. If we take $e=1,$ then any complex number $c=a+ib$ is a multiple of $e:$ $c=c1=ce$ with the scaling coefficient $c.$ Thus, $C$ is a one-dimensional space in this sense. On the other hand, if only multiplication by real numbers is allowed, then we can take $e_1=1,$ $e_2=i$ as a basis and then $c=ae_1+be_2$ and $C$ is two-dimensional. To avoid confusion, just use scaling by the right numbers.

The scalar product in $R^n$ is given by $x\cdot y=\sum x_iy_i$ and in $C^n$ by $x\cdot y=\sum x_i\bar{y}_i.$ As a result, for the second scalar product we have $x\cdot (c_1y+c_2z)=\bar{c}_1x\cdot y+\bar{c}_2x\cdot z$ for complex $c_1,c_2$ (some people call this antilinearity, to distinguish it from linearity $x\cdot (ay+bz)=ax\cdot y+bx\cdot z$ for real $a,b$ ).

Definition 1. For a matrix $A$ with possibly complex entries we denote $A'=\overline{A^T}.$ The matrix $A'$ is called an adjoint or a conjugate of $A.$

Exercise 1. Prove that $(Ax)\cdot y=x\cdot(A'y),$ for any $x,y\in C^n.$

Proof. For complex numbers we have $\overline{\bar{c}}=c,$ $\overline{c_1c_2}=\bar{c}_1\bar{c}_2.$ Therefore

$(Ax)\cdot y=(Ax)^T\bar{y}=x^TA^T\bar{y}=x^T\overline{\overline{(A^T)}}\bar{y}=x^T\overline{(\overline{A^T}y)}=x\cdot(A'y).$

Thus, when considering matrices in $C^n,$ conjugation should be used instead of transposition. In particular, instead of symmetry $A=A^T$ the equation $A=A'$ should be used. Matrices satisfying the last equation are called self-adjoint. The theory of self-adjoint matrices in $C^n$ is very similar to that of symmetric matrices in $R^n.$ Keeping in mind two applications (regression analysis and optimization), we consider only square matrices with real entries. Even in this case one is forced to work with $C^n$ from time to time because, in general, eigenvalues can be complex numbers.

General properties of symmetric matrices

$A$ is assumed a square matrix with real entries. When we extend $A$ from $R^n$ to $C^n,$ $Ax$ is defined by the same expression as before but $x$ is allowed to be from $C^n$ and the scalar product in $R^n$ is replaced by the scalar product in $C^n.$ The extension is denoted $A_C.$

Exercise 2. If $A$ is symmetric, then all eigenvalues of $A_C$ are real.

Proof. Suppose $\lambda$ is an eigenvalue of $A_C.$ Using Exercise 1 and the symmetry of $A$ we have

$\lambda x\cdot x=(Ax)\cdot x=x\cdot(Ax)=x\cdot(\lambda x)=\bar{\lambda}x\cdot x.$

Since $x\cdot x=\|x\|^2>0,$ we have $\lambda =\bar{\lambda}.$ This shows that $\lambda$ is real.

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

Proof. We know that $A_C$ has at least one complex eigenvalue $\lambda$ . By Exercise 2, this eigenvalue must be real. Thus, we have $Ax=\lambda x$ with some nonzero $x\in C^n.$ Separating real and imaginary parts of $x,$ we have $x=u+iv,$ $Au=\lambda u,$ $Av=\lambda v$ with some $u,v\in R^n.$ At least one of $u,v$ is not zero. Thus a real eigenvector exists.

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

Definition 2. A subspace $L$ is called an invariant subspace of $A$ if $AL\subseteq L.$

Example 2. If $x$ is an eigenvector of $A,$ then the subspace $L$ spanned by $x$ is an invariant subspace of $A.$ This is because $x\in L$ implies $Ax=\lambda x\in L.$

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

Proof. By the definition of an invariant subspace, the restriction of $A$ to $L$ defined by $A_Lx=Ax,$ $x\in L,$ acts from $L$ to $L$ . By Exercise 3, applied to $A_L,$ it has an eigenvector in $L$ , which is also an eigenvector of $A.$

Exercise 5. a) If $\lambda$ is an (real) eigenvalue of $A_R,$ then it is an eigenvalue of $A_C.$ b) If $\lambda$ is a real eigenvalue of $A_C,$ then it is an eigenvalue of $A_R.$ This is summarized as $\sigma(A_R)=\sigma (A_C)\cap R,$ see the spectrum notation.