**Module leader**: Christian Böhning --- # 1. Jordan Canonical Form ###### 1.4 The Cayley-Hamilton theorem See [[Cayley-Hamilton theorem]]. ###### 1.5 Calculating the minimal polynomial **Theorem 1.5.2**: TBC ###### 1.6/7/8/9/10/11 Jordan canonical form **Lemma 1.6.3** [[Jordan chain is linearly independent]]. **Theorem 1.7.1** [[Existence and uniqueness of Jordan basis for endomorphism of finite-dimensional complex vector space]] **Theorem 1.7.4:** [[Characterization of characteristic and minimal polynomials by Jordan canonical form]]. **General algorithm for JCF**: TBC. **Theorem 1.11.3**: [[Spectral characterisation of similar matrices]]. # 2. Functions of Matrices See [[Functions of Jordan blocks]]. # 3. Bilinear Maps and Quadratic Forms ###### 3.1/2 Bilinear maps Definition 3.1.1. [[Bilinearity]] Theorem 3.2.1/2. [[Matrix representations of bilinear map]]. Proposition 3.2.6. [[Normal form of bilinear form on vector space whose characteristic is not two]]. ###### 3.3/4 Quadratic forms Definition 3.3.1. [[Quadratic forms]]. Theorem 3.4.1. [[Symmetric matrix is congruent diagonal matrix]]. Theorem 3.4.4. [[Sylvester's law of inertia]]. ###### 3.5 Gram-Schmidt We define a [[Euclidean spaces|Euclidean space]] as real vector space $V$ together with a symmetric bilinear form $\tau:V\times V \to \mathbb{R}$ such that $\tau(v,v)>0$ for all $v\in V$ (i.e. $\tau$ is positive definite). **Theorem 3.5.3.** [[Gram-Schmidt orthogonalisation in Euclidean space]]. ###### 3.6 Orthogonal transformations 3.6.1/2/3. [[Orthogonal endomorphisms of Euclidean spaces]]. 3.6.6 [[QR decomposition]]. ###### 3.7/8/9 Orthonormal basis for quadratic forms, SVD We define adjoints and self-adjoint endomorphisms of Euclidean spaces. **Theorem 3.7.3**. [[Spectral theorem]]. **Theorem 3.9.1**. [[Singular value decomposition]]. # 4. Duality, quotients, tensors and all that See [[Dual of vector space]] See [[Dual of linear map]]. See [[Quotient vector spaces]]. See [[Tensor product of vector spaces]]. See [[Tensor algebra of vector space]]. See [[Symmetric algebra of vector space]] and [[Exterior algebra of vector space]].