Basic knowledge and formulas of matrices (transpose, inverse, trace, determinant)

References: MatrixCookBook(Version 2012) Chapter1

Chapter1: Basics

1 Basics

Note： A H {A^H} AHIt's ATransposed and complex conjugated matrix (Hermitian), Namely transpose the complex conjugate matrix.

1.1 Trace of matrix (Trace)

formula(11)Shows that the trace of the matrix is ​​the sum of the main diagonal elements.
formula(12)Shows that the trace of the matrix is ​​the sum of the eigenvalues ​​of the matrix.
formula(13)Shows that the trace of the matrix is ​​equal to the trace of its transposed matrix.
formula(14)showABThe trace is equal toBAOf traces.
formula(15)showA+BThe trace is equal toATrace plusBOf traces.
formula(16)showABCThe trace is equal toBCAThe trace is equal toCABOf traces.
formula(17)Indicate anx1Vector ofa，aMultiply the transpose ofaThe resulting constant is equal toaMultiply byaThe trace of the transposed matrix.

1.2 Determinant (Determinant)

premise: The A here isnxnmatrix.
formula(18)Shows that the determinant of the matrix is ​​equal to the product of the eigenvalues.
formula(19)showcAThe determinant of is equal toADeterminant c n {c^n} cnTimes.
formula(20)Shows that the determinant of the matrix is ​​equal to the determinant of its transposed matrix.
formula(21)Show matrixABThe determinant of is equal to the matrixAMultiply the determinant of the matrixBThe determinant.
formula(22)Show matrix A − 1 {A^{-1}} A−1The determinant of is equal to the matrixAReciprocal.
formula(23)Show matrix A n {A^n} AnThe determinant of is equal to the matrixAN to the power of the determinant.
formula(24)Show ifuwithvYesnx1Vector, then I + u v T {I+uv^T} I+uvTThe determinant of is equal to 1 + u T v {1+u^Tv} 1+uTvValue.
formula(25)Show ifAYes2x2matrix,I+AThe determinant of is equal to 1 + d e t ( A ) + T r ( A ) {1+det(A)+Tr(A)} 1+det(A)+Tr(A), That is, the determinant of 1+A + the trace of A.
formula(26)Show ifAYes3x3matrix,I+AThe determinant of is equal to 1 + d e t ( A ) + T r ( A ) + 1 2 T r ( A ) 2 − 1 2 T r ( A 2 ) {1+det(A)+Tr(A)+\frac{1}{2}Tr(A)^2-\frac{1}{2}Tr(A^2)} 1+det(A)+Tr(A)+21​Tr(A)2−21​Tr(A2)。
formula(27)No table.
formula(28)Means for small disturbances ε \varepsilon ε,can ε A \varepsilon A εAApproximately processed as 2x2

1.3 Special case: 2x2 matrix

2x2The matrix has the above properties and conclusions.