Chapter 2 Real Symmetric Matrices 2.1 Special properties of real symmetric matrices A matrixA M n(C) (orM n(R)) is diagonalizable if it is similar to a diagonal matrix. If this ∈ happens, it means that there is a basis ofC n with respect to which the linear transformation ofC n defined by left multiplication byA has a diagonal matrix. Every element of such a basis is simply multiplied by a scalar when it is multiplied byA, which means exactly that the basis consists of eigenvectors ofA. n Lemma 2.1.1. A matrixA M n(C) is diagonalizable if and only ifC possesses a basis consisting of ∈ eigenvectors ofA. 1 1 Not all matrices are diagonalizable. For exampleA= is not. To see this note that 1 0 1 � � (occurring twice) is the only eigenvalue ofA, but that all eigenvectors ofA are scalar multiples of 1 2 2 0 , soC (orR ) does not contain a basis consisting of eigenvectors ofA, andA is not similar to a diagonal matrix. � � We note that a matrix can fail to be diagonalizable only if it has repeated eigenvalues, as the following lemma shows. Lemma 2.1.2. LetA M n(R) and suppose thatA has distinct eigenvaluesλ 1,...,λ n inC. ThenA is ∈ diagonalizable. Proof. Letv i be an eigenvector ofA corresponding toλ i. We will show thatS={v 1,...,v n} is a linearly independent set. Sincev 1 is not the zero vector, we know that{v 1} is linearly independent. IfS is linearly dependent, letk be the least for which{v 1,...,v k} is linearly dependent. This means that{v 1,...,v k−1} is a linearly independent set and v =a v + +a v k 1 1 ··· k−1 k−1 for somea i C, not all zero. Multiplying this equation on the left separately byA and byλk ∈ gives λkvk =a 1λ1v1 +a 2λ2v2 +a k−1λk−1vk−1 a1λkv1 +a 2λkv2 +a k−1λkvk−1 = 0=a (λ −λ )v +a (λ −λ ) + +a (λ −λ )v . ⇒ 1 1 k 1 2 2 k ··· k k−1 k k−1 Since the complex numbersλ i −λ k are non-zero fori= 1, . ,k− 1 and at least one of thesea i is non-zero, the above is an expression for the zero vector as a nontrivial linear combination of v1,...,v k−1, contrary to the choice ofk. We conclude thatS is linearly independent and hence that it is a basis ofC n. 13 Definition 2.1.3. A matrixA M n(R) is symmetric if it is equal to its transpose, i.e. ifA ij =A ji for ∈ alli andj. Symmetric matrices arise naturally in various contexts, including as adjacency matrices of undirected graphs. Fortunately they have lots of nice properties. To explore some of these we need a slightly more general concept, that of a complex Hermitian matrix. Definition 2.1.4. LetA M n(C). The Hermitian transpose, or conjugate transpose ofA is the ∈ matrixA ∗ obtained by taking the transpose ofA and then taking the complex conjugate of each entry. The matrixA is said to be Hermitian ifA=A ∗. Notes 2+i4−i 2−i3 1. Example: IfA= , thenA ∗ = 3 3−i 4+i3+i � � � � 2. The Hermitian transpose ofA is equal to its (ordinary) transpose if and only ifA M n(R). ∈ In some contexts the Hermitian transpose is the appropriate analogue inC of the concept of transpose of a real matrix. 3. IfA M n(C), then the trace of the productA ∗A is the sum of all the entries ofA, each ∈ multiplied by its own complex conjugate (check this). This is a non-negative real number T and it is zero only ifA= 0. In particular, ifA M n(R), then trace(A A) is the sum of the ∈ squares of all the entries ofA. 4. Suppose thatA andB are two matrices for which the productAB exists. Then(AB) T = T T B A and(AB) ∗ =B ∗A∗ (it is routine but worthwile to prove these statements). In particu- lar, ifA is any matrix at all, then T T T T T T (A A) =A (A ) =A A, and(A ∗A)∗ =A ∗(A∗)∗ =A ∗A, T T soA A andA ∗A are respectively symmetric and Hermitian (so areAA andAA ∗). The following theorem is the start of the story of what makes real symmetric matrices so special. Theorem 2.1.5. The eigenvalues of a real symmetric matrix are all real. Proof. We will prove the stronger statement that the eigenvalues of a complex Hermitian matrix are all real. LetA be a Hermitian matrix inM n(C) and letλ be an eigenvalue ofA with corre- n sponding eigenvectorv. Soλ C andv is a non-zero vector inC . Look at the productv ∗Av. ∈ This is a complex number. v∗Av=v ∗λv=λv ∗v. The expressionv ∗v is a positive real number, since it is the sum of the expressions ¯ivvi over all entriesv i ofv. We have not yet used the fact thatA ∗ =A. Now look at the Hermitian transpose of the matrix productv ∗Av. (v∗Av)∗ =v ∗A∗(v∗)∗ =v ∗Av. This is saying thatv ∗Av is a complex number that is equal to its own Hermitian transpose, i.e. equal to its own complex conjugate. This means exactly thatv ∗Av R. ∈ We also know thatv ∗Av=λv ∗v, and sincev ∗v is a non-zero real number, this means that λ R. ∈ So the eigenvalues of a real symmetric matrix are real numbers. This means in particular that the eigenvalues of the adjacency matrix of an undirected graph are real numbers, they can be arranged in order and we can ask questions about (for example) the greatest eigenvalue, the least eigenvalue, etc. 14 Another concept that is often mentioned in connection with real symmetric matrices is that T of positive definiteness. We mentioned above that ifA M m n(R), thenA A is a symmetric ∈ × matrix. However not every symmetric matrix has the formA T A, since for example the entries on the main diagonal ofA T A do not. It turns out that those symmetric matrices that have the form AT A (even for a non-squareA) can be characterized in another way. Definition 2.1.6. LetA be a symmetric matrix inM n(R). ThenA is called positive semidefinite (PSD) ifv T Av 0 for allv R n. In addition, ifv T Av is strictly positive wheneverv=0, thenA is called � ∈ � positive definite (PD). Notes 1. The identity matrixI n is the classical example of a positive definite symmetric matrix, since n T T for anyv R ,v Inv=v v=v v 0, andv v= 0 only ifv is the zero vector. ∈ · � · 1 2 2. The matrix is an example of a matrix that is not positive semidefinite, since 2 1 � � 1 2 −1 −1 1 =−2. 2 1 1 � � � � � � So positive (semi)definite is not the same thing as positive - a symmetric matrix can have all of its entries positive and still fail to be positive (semi)definite. 3. A symmetric matrix can have negative entries and still be positive definite, for example the 1−1 matrixA= is SPD (symmetric positive definite). To see this observe that for −1 2 � � real numbersa andb we have 1−1 a a b =a 2 − ab− ab+2b 2 = (a−b) 2 +b 2. −1 2 b � � � � � � Since(a−b) 2 +b 2 cannot be negative and is zero only if botha andb are equal to zero, the matrixA is positive definite. The importance of the concept of positive definiteness is not really obvious atfirst glance, it takes a little bit of discussion. We will defer this discussion for now, and mention two observations related to positive (semi)definiteness that have a connection to spectral graph theory. Lemma 2.1.7. The eigenvalues of a real symmetric positive semidefinite matrix are non-negative (positive if positive definite). Proof. Letλ be an eigenvalue of the real symmetric positive semidefinite matrixA, and letv R n ∈ be a corresponding eigenvector. Then T T T 0�v Av=v λv=λv v. Thusλ is nonnegative sincev T v is a positive real number. Lemma 2.1.8. LetB M n m(R) for some positive integersm andn. Then the symmetric matrix T ∈ × A=BB inM n(R) is positive semidefinite. Proof. Letu R n. Then ∈ uT Au=u T BBT u=(u T B)(BT u) = (BT u)T (BT u) = (BT u) (B T u) 0, · � soA is positive semidefinite. The next section will contain a more detailed discussion of positive (semi)definiteness, includ- ing the converses of the two statements above. First we digress to look at an application of what we know so far to spectral graph theory. 15 Definition 2.1.9. LetG be a graph. The line graph ofG, denoted byL(G), has a vertex for every edge ofG, and two vertices ofL(G) are adjacent if and only if their corresponding edges inG share an incident vertex. Example 2.1.10.K 4 (left) and its line graph (right). Choose an edge ofK 4. Since each of its incident vertices has degree 3, there are four other edges with which it shares a vertex. So the vertex that represents it inL(K 4) has degree 4. In general, if a graphG is regular of degreek, thenL(G) will be regular of degree 2k− 2. For any graphG, a vertex of degreed inG corresponds to a copy of the complete graphK d withinL(G). Not every graph can be a line graph.