Select Page

* Square matrix: If the matrix has same number of columns as rows

* Symmetric matrix: A square matrix that is equal to its transpose. $latex \mathbf{A} = \mathbf{A}^T * Diagonal matrix: A matrix where all non-diagonal entries are zero$latex $\mathbf{A}_{i,j} = 0, \forall i \ne j$. Square diagonal matrices are denoted as $latex \text{diag}(v)$. But note that diagonal matrices are not required to be square.

* Identity matrix: A diagonal matrix where all diagonal entries are 1

* Inverse matrix: An inverse of a matrix is another matrix such that their product is an identity matrix. $latex \mathbf{A}^{-1}\mathbf{A} = \mathbf{A}\mathbf{A}^{-1} = 1$

* Inverse of diagonal matrix: Is easy to compute: $latex \text{diag}(v)^{-1} = \text{diag}([1/v_1,\ldots,1/v_n])^T$

* Unit vector: A vector with unit norm $latex ||\mathbf{x}\||_2 = 1$

* Orthogonal vectors: If $latex \mathbf{x}^T\mathbf{y} = 0$, then $latex \mathbf{x}$ and $latex \mathbf{y}$ are considered orthogonal. If their norms are non-zero, then it is the case that the angle between them is $latex 90^{\circ}$. Why? Because a product of vectors can be represented as $latex \mathbf{x}^T\mathbf{y} = |\mathbf{x}||\mathbf{y}|\cos\theta$, where $latex \theta$ is the angle between the two vectors.

* Orthonormal vectors: If two orthogonal vectors are also unit vectors, then they are orthonormal

* Orthogonal Matrix: A square matrix whose rows are mutually orthornomal and the columns are mutually orthonormal. Therefore $latex \mathbf{A}^T\mathbf{A} = \mathbf{A}\mathbf{A}^T = \mathbf{I}$

* Positive definite Matrix: A square symmetric matrix $latex \mathbf{A}$ such that for any non-zero vector $latex \mathbf{x}$, $latex \mathbf{x}^T\mathbf{A}\mathbf{x} > 0$. It is sometimes denoted as $latex \mathbf{A} > 0$

* Positive semi-definite Matrix: A square symmetric matrix $latex \mathbf{A}$ such that for any non-zero vector $latex \mathbf{x}$, $latex \mathbf{x}^T\mathbf{A}\mathbf{x} \ge 0$. It is sometimes denoted as $latex \mathbf{A} \ge 0$.