Linear Econometric for Finance: notes
. Hopkins University
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Chapter 0 Empirical finance
Quick review of important facts about matrices
• Identity matrix. A square matrix with ones on the diagonal and zeros everywhere else.
• Transposition. Let aij denote the row i, column j element of a matrix A : A = [aij ]. The transpose of A, denoted by A′, is given by A′ = [aji].
• Important properties:
(AB)′ = B′A′ (A+B)′ = A′+B′
• Symmetric matrices. A square matrix such that A = A′ is said to be symmetric.
• Trace. The trace of a square matrix is the sum of the elements along the diagonal.
• Important property:
tr(AB) = tr(BA)
• Determinant. The determinant of a 2 × 2 square matrix A, denoted by |A|, is the
|A| = a11a22 − a12a21
• One could of course define determinants for more general matrices than a simple
2 × 2 square matrix.
• Important properties:
1. The determinant of the identity matrix is 1.
2. For an n×n matrix A,
|σA| = σn|A| 1
Chapter 0 Empirical finance • Inverse. If the determinant of a square matrix exists, then its inverse exists and is
such that A × A−1 = I (the identity matrix). In the bi-variate case: 1 a−a
A−1= 2212. a11a22 − a12a21 −a21 a11
• Eigenvalues and eigenvectors.
• Suppose that an n×n matrix A, a nonzero n×1 vector x, and a scalar λ are related
• Then, x is called the eigenvector of A and λ is called the eigenvalue of A.
• Jordan decomposition. Every symmetric matrix A can be written as BΛB′, where Λ is a matrix which contains the eigenvalues of A on the diagonal (and zeros ev- erywhere else) and B is an orthogonal matrix consisting of the eigenvectors of A.
• A square orthogonal matrix is such that BB′ = I and B−1 = B′.
• Idempotent matrices. A square matrix A such that AA = A is called idempotent.
• Important property: The eigenvalues of an idempotent matrix are either 1 or zero.
• Positive semidefinite. An n × n real symmetric matrix A is said to be positive semidefinite if, for any real n × 1 vector x, x′Ax ≥ 0.
• Important property: The eigenvalues of a positive semidefinite matrix are either zero or positive.
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