Part 1 Definitions, Basic Concepts, and Matrix Operationsġ Vector Spaces, Subspaces, and Linear Transformationsġ.3.1 The Range and Null Spaces of a Linear Transformationģ.1.1 First- and Second-Order Determinantsģ.3.4 Adding Multiples of a Row (Column) to a Row (Column)ģ.4.4 Adding a Row to a Multiple of a Rowģ.9 A Graphical Representation of a DeterminantĤ.13 Rank of a Matrix-Some Preliminary ResultsĤ.14 The Number of LIN Rows and Columns in a MatrixĤ.15 Determination of The Rank of a MatrixĤ.18.3 Matrices of Full Row (Column) Rankĥ.5 Parameterization of Orthogonal MatricesĦ.2.2 Eigenvalues of a Scalar-by-Matrix ProductĦ.5.4 Rank Equals Number of Nonzero Eigenvalues for a Symmetric MatrixĦ.6 Eigenvalues of orthogonal and Idempotent MatricesĦ.6.1 Eigenvalues of Symmetric Positive Definite and Positive Semidefinite MatricesĦ.7 Eigenvalues of Direct Products and Direct Sums of Matricesħ.1.1 The Number of Nonzero Eigenvalues Never Exceeds Rankħ.1.4 All Symmetric Matrices Are Diagonableħ.2.2 Simultaneous Diagonalization of Two Symmetric MatricesĨ.2.1 Derivation Using the Singular-Value DecompositionĨ.2.2 Derivation Based on Knowing the Rankĩ.2 Iterative Solution of Nonlinear Equationsĩ.5.3 Derivative of a Matrix with Respect to Another Matrixĩ.6 Matrices With Elements That Are Complex Numbers The Introduction of Matrices Into Statistics
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