Scalar: A scalar is an element of a filed which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.
Positive Definite: A symmetric $n\times n$ real matrix $M$ is said to be postive definite if the scalar $z^TMz$ is strictly positive for every non-zero column vector $z$ of $n$ real numbere.
Positive semi-definite matrices are defined similarly, except that the above scalars $z^TMz$ must be positive or zero.
Trace: The trace of an $n\times n$ square matrix A is defined to be the sum of the elements on the main diagonal of A.
Block Diagonal Matrix: A block diagonal matrix, is a square diagonal matrix in which the diagonal elements are square matrices of any size (possibly even $1\times1$), and the off-diagonal elements are 0.
Direct Sum: Let $U,W$ be subspaces of $V$. Then V is said to be the direct sum of $U$ and $W$, and we write $V = U \oplus W$, if $V=U+W$ and $U\cap W = {0}$
Rank: The rank of a matrix A is the dimension of the vector space generated by its columns. This corresponds to the maximal number of linearly independent columns of A.