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APPENDIX A. A primer on linear matrix algebra
Note: this is a simplified presentation for finite-dimensional real vector spaces. For more general results and rigorous mathematical definitions, refer to mathematical textbooks.
Matrix. A matrix





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A matrix for which

Diagonal. The diagonal of a square




Transpose. The transpose of a







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Symmetry. A square matrix is symmetric if it is equal to its transpose, i.e.




Scalar multiplication. A






Matrix sum. The sum of two







Matrix product. The product between an







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The product is not defined if the number of columns in











Matrix inverse. A square






Trace. The trace of a square



Useful properties.
(A, B, C are assumed to be such that the operations below have a meaning)
The transposition is linear:

Transpose of a product:

Inverse of a product:

Inverse of a transpose:

Associativity of the product:

Diagonal matrices: their products and inverses are diagonal, with coefficients given respectively by the products and inverses of the diagonals of the operands.
Symmetric matrices: the symmetry is conserved by scalar multiplication, sum and inversion, but not by the product (in general).
The trace is linear:

Trace of a transpose:

Trace of a product:

Trace and basis change:


Positive definite matrices. A symmetric matrix




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