Rank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth col
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1 Review of Linear Algebra { E18 Hanout Vectors an Their Inner Proucts Let X an Y be two vectors: an Their inner prouct is ene as X =[x1; ;x n ] T Y =[y1; ;y n ] T (X; Y ) = X T Y = x k y k k=1 where T an represent transpose an complex conjugate, respectively. The norm (magnitue, length) of a vector X is ene as kxk =(X; X) 1= = vu u t n X jx k j k=1 where jxj represents the absolute value of x (real or complex). X is normalize if kxk =1. Two vectors X an Y are orthogonal to each other i their inner prouct is zero. For normalize orthogonal vectors, we have ( (X; Y )= 1 if X = Y XY = 0 if X = Y 1
2 Rank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth column vector an a1j aj ::: a nj [a i1 a i a in ] is the ith row vector. The n rows span the row space of A an the n columns span the column space of A. The imensions of these two spaces are the same an calle the rank of A: R = rank(a) N The eterminant of A is enote by an we have et(a) =jaj jabj = jaj jbj rank(a) <N i et(a) = 0. The trace of A is ene as the sum of its iagonal elements: tr(a) = a ii
3 The transpose of a matrix A, enote by A T, is obtaine by switching the positions of elements a ij an a ji for all i; j f1; ;ng. In other wors, the ith column of A becomes the ith row of A T, or equivalently, the ith row of A becomes the ith column of A T : A T =[A1 A n ] T = A T 1 :: :: A T n where vector A i is the ith column of A an its transpose A T i of A T. For any two matrices A an B, we have is the ith row (AB) T = B T A T If AB = BA = I, where I is an ientity matrix: I = iag[1; ; 1] = then B = A,1 is the inverse of A. A,1 exists i et(a) = 0, i.e., rank(a) = N. For any two matrices A an B, we have (AB),1 = B,1 A,1 an (A,1 ) T =(A T ),1
4 Hermitian Matrix an Unitary Matrix A is a Hermitian matrix i A T = A. When a Hermitian matrix A is real (A = A), it becomes a symmetric matrix, A T = A. A is a unitary matrix i A T A = I, i.e.,a T = A,1. When a unitary matrix A is real (A = A), it becomes an orthogonal matrix, A T = A,1. The columns (or rows) of a unitary matrix A are orthonormal, i.e. they are both orthogonal an normalize, i.e., ( (A i ;A j )= 1 if i = j ij = 0 if i = j where A i an A j represent the ith an jth columns of A, respectively. As we will see later, any Hermitian matrix A can be converte to a iagonal matrix (or iagonalize) by a particular unitary matrix : T A= where is a iagonal matrix, i.e., all its o iagonal elements are 0.
5 Unitary Transforms For any given unitary matrix A =[A1 A A n ] T,aunitary transform of a vector X =[x1;x; ;x n ] T can be ene as 8 >< >: Y = A T X = A T 1 ::: A T n X X = AY =[A1;A; ; A n ] Y where Y =[y1;y; ;y n ] T is another vector. The rst equation of the unitary transform is the forwar transform an the ith component of Y can be written as: y i = A T i X =(A i ;X) y i is the inner prouct of X an the ith column vector A i of A, i.e., the projection of X on the ith vector A i. The secon equation is the inverse transform an can be written as X = y i A i i.e., vector X is represente as a linear combination (weighte sum) of the n column vectors A i ;A; ;A n of the transform matrix A. In other wors, X is represente as a vector (or a point) in the n-imensional space spanne by the n orthonormal column vectors A1;A; ;A n. Each of the n coorinates (y1;y; ;y n ) of this vector is its projection on the irection specie by the corresponing column vector of A. Specially when A = I, we have X = y i A i = x i I i where I i = [0; ; 0; 1; 0; ; 0] T is the ith column of the ientity matrix I with the ith element equal 1 an all other 0.
6 Unitary transform oes not change a vector's norm: ky k = Y T Y =(A T X) T (A T X)=X T AA T X = X T X = kxk as AA T = I. In other wors, the length of a vector is always the same in ierent coorinate systems. The geometric interpretation of any unitary transform Y = AX is to rotate a vector about the origin [0; ; 0] T (rotation oes not change the vector's length), or equivalently, to represent the same vector X by the coorinates Y in a ierent coorinate system. If X is interprete as a signal, then kxk can be interprete as the total energy or information containe in the signal which is preserve by uring any unitary transform. However, some other features of the signal may be change, e.g., the signal may be ecorrelate after the transform, which is esirable in many applications.
7 Eigenvalues an Eigenvectors For any matrix A, if there exist a vector anavalue such that A = then an are calle the eigenvalue an eigenvector of matrix A, respectively. To obtain, rewrite the above equation as (I, A) =0 which is a homogeneous equation system. To n its non-zero solution for, we require ji, Aj =0 Solving this nth orer equation of, we get n eigenvalues f1; ; n g. Substituting each i back into the equation system, we get the corresponing eigenvector i. We now have A[1; ; n ]=[11; ; n n ]=[1; ; n ] or in a more compact form, or where an A =,1 A= =[1; ; n ] =iag[1; ; n ] n
8 an The trace an eterminant of A can be obtaine from its eigenvalues tr(a) = k=1 ny et(a) = k=1 A T has the same eigenvalues an eigenvectors as A. A m has the same eigenvectors as A, but its eigenvalues are f m 1 ; ;m n g, where m is a positive integer. This is also true for m =,1, i.e., the eigenvalues of A,1 are f1=1; ; 1= n g. If A is Hermitian (symmetric if A is real), all the i 's are real an all eigenvectors i 's are orthogonal: k ( i ; j )= ij If all i 's are normalize, matrix is unitary (orthogonal if A is real):,1 = T k an we have,1 A= T A= 8
9 Positive Denite Matrix A real symmetric matrix A is positive enite, enote by A > 0, i the quaratic form X T AX is greater than zero: X T AX > 0 for any X =[x1; ;x n ] T (x i 's are not all zero). A>0 i all its eigenvalues are greater than zero: i > 0; i =(1; ;n) As the eigenvalues of A,1 are 1= i ; i = (1; ;n), we have A > 0 i A,1 > 0. 9
10 Vector Dierentiation A vector ierentiation operator is ene as ; ] n which can be applie to any scalar function f(x) tonits erivative with respect to f(x) =[@f ; ; ] n Vector ierentiation has the following properties: X (BT X)= X (X T B)=B X (X T X)=X X (X T AX) =AX (if A T = A) To prove the thir one, consier the kth element of k (X T AX) a ij x i x j = a ik x i + a kj x j = a ik x k j=1 j=1 for (k =1; ;n). Note that here we have use the assumption that a ik = a ki, i.e., A T = A. Putting all n elements in vector form, we have the above. When A = I, we have X (X T X)=X You can compare these results with the familiar erivatives in the scalar case: x (ax )=ax 10
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