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Orthogonal matrix is a real square matrix whose product, with its transpose, gives an identity matrix. When two vectors are said to be orthogonal, it means that they are perpendicular to each other. When these vectors are represented in matrix form, their product gives a square matrix. A square matrix is said to be orthogonal when it comprises real elements and its transpose is equal to its inverse. In other words, when the product of the real square matrix and its transpose is equal to an identity matrix, the real square matrix is said to be an orthogonal matrix.
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Key Terms: Orthogonal matrix, Identity matrix, orthogonally diagonalizable matrix, orthogonal matrix 2x2, square matrix, vectors, rows, columns
What is a Matrix?
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A matrix is a rectangular array of numbers arranged in rows and columns. The standard form of the matrix is:
Representation of Matrix
It has m columns and n rows. aij represents an element of the matrix whose row is i and column is j.
The video below explains this:
Matrices Detailed Video Explanation:
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| Related Articles | ||
|---|---|---|
| Matrices | Types of Matrices | Operations on Matrices |
| Transpose of a Matrix | Symmetric and Skew Symmetric Matrix | Invertible Matrix |
| Singular Matrix | Determinants | Properties of Determinants |
Other Types of Matrix
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The other types of matrices are discussed below.
Square Matrix
In a square matrix, the number of rows and columns are equal. So, when m = n for the standard matrix, it is a square matrix.
\(\text{ Square Matrix M} = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix}\)
Square matrix with 3 rows and 3 columns
Transpose of a Matrix
To get a transpose of a matrix, flip the matrix over its diagonal. So basically, when you switch the rows with the columns of a matrix and its columns with rows, you get the transpose of a matrix. It is denoted by AT.
\(\begin{bmatrix} 6 & 4 & 24 \\ 1 & -9 & 8 \end{bmatrix} ^T = \begin{bmatrix} 6 & 1 \\ 4 & -9 \\ 24 & 8 \end{bmatrix}\)
Transpose of a Matrix
Identity Matrix
A square matrix, with 1s in the main diagonal position and 0s elsewhere, is called an identity matrix. It is denoted by I or In.
Identity Matrix
Determinant of a Matrix
The determinant of a matrix is the scalar value that is the function of the elements of a square matrix. It is denoted by det A, det(A) or |A|.
Matrix: Determinant:
\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) det(A) = |A| = \(\begin{vmatrix} a & b\\ c & d \end{vmatrix}\)
Determinant of a Matrix
Inverse of a Matrix
The product of a matrix and its inverse gives an identity matrix. The inverse of matrix A is denoted by A-1. The inverse of a matrix exists only for square matrices with non-zero determinant values.
A-1 = adj A / |A|, where |A| ≠ 0
Inverse of a Matrix
\(A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)
Symmetric Matrix
A symmetric matrix is a square matrix whose transpose is equal to the matrix itself.
A = AT
Here, if you check the AT, it will be the same as A. So, A is a symmetric matrix.
\(\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 8 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 8 \end{bmatrix}^T\)
Symmetric matrix
What is an Orthogonal Matrix?
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A square matrix is said to be orthogonal when it comprises real elements and its transpose is equal to its inverse. In other words, when the product of the real square matrix and its transpose is equal to an identity matrix, the real square matrix is said to be an orthogonal matrix.
Let A be the square matrix, AT is the transpose of A and A-1 is the inverse of A.
If
AT = A-1
then
AAT = ATA = I
Here, I is the identity matrix.
Properties of an Orthogonal Matrix
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- An orthogonal matrix is a real square matrix. (This means that all the elements of an orthogonal matrix are real numbers.)
- All orthogonal matrices are invertible.
- The determinant of an orthogonal matrix is +1 or -1.
- All orthogonal matrices are square matrices, but all square matrices are not orthogonal matrices.
- The inverse of an orthogonal matrix is also an orthogonal matrix.
\(P^T QR = \begin{bmatrix} R_1 & {} & {} \\ {} & \ddots & {} \\ {} & {} &R_k \end{bmatrix} \text{(n even) ,} P^T QP = \begin{bmatrix} R_1 & {} & {} & {} \\ {} & \ddots & {} & {} \\ {} & {} & R_k & {} \\ {} & {} & {} & 1 \end{bmatrix} (\text{n odd})\)
- All identity matrices are orthogonal matrices.
- The product of orthogonal matrices is an orthogonal matrix.
- All orthogonal matrices of the order n x n are collectively known as an orthogonal group, which is denoted by O.
- The transpose of an orthogonal matrix is also an orthogonal matrix.
- All orthogonal matrices are symmetric.
- For an orthogonal matrix, its inverse and transpose are equal.
Dot Product of Orthogonal Matrices
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When two vectors are perpendicular to each other, their dot product is zero. So, the dot product of two orthogonal matrices is zero, as their vectors are perpendicular to each other.
Things To Remember
- A real square matrix whose inverse is equal to its transpose is called an orthogonal matrix.
AT = A-1
- For an orthogonal matrix, the product of the matrix and its transpose are equal to an identity matrix.
AAT = ATA = I
- The determinant of an orthogonal matrix is +1 or -1.
- All orthogonal matrices are symmetric and invertible.
- Inverse of an orthogonal matrix is also an orthogonal matrix. And transpose of an orthogonal matrix is also an orthogonal matrix.
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Sample Questions
Ques. Check if A is an orthogonal matrix. (2 marks)
A = [6 2 3 9]
Ans. The determinant of an orthogonal matrix is +1 or -1.
det A = (6 x 9) – (2 x 3) = 54 – 6 = 48
Hence, A is not an orthogonal matrix.
Ques. Verify that A = [cos x sin x - sin x cos x] is an orthogonal matrix. (2 marks)
Ans. A = [cos x sin x -sin x cos x]
AT = [cos x -sin x sin x cos x]
AAT= [(cos x)(cos x)+(sin x)(sin x)(cos x)(-sin x)+(sin x)(cos x) (-sin x)(cos x )+(cos x) (sin x ) (-sin x) (-sin x) +(cos x)(cos x) ]
= [1 0 0 1] = I2
AAT = I
Ques. What is a Matrix Polynomial? (2 marks)
Ans. If f (x) = a0xn + a1xn-1 + a2Xn-2 + ……… + anx then we define a matrix polynomial f(A) = a0An + a1An-1 + a2An-2 + ……… + an? where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomial f (x).
Ques. What is a Transpose of a Matrix? (2 marks)
Ans. The Transpose Of A Matrix: (Changing rows & columns)
Let A be any matrix. Then, A = aij of order m × n
⇒ AT or A′ = [ aij ] for 1 ≤ i ≤ n & 1 ≤ j ≤ m of order n × m
Properties of Transpose of a Matrix:
If At & Bt denote the transpose of A and B ,
- (A ± B)t = At ± B?; note that A & B have the same order.
- (AB)t = BtAt A & B are conformable for matrix product AB.
- (At)t = A
- (kA)t = kAt k is a scalar.
- General : (A1, A2, …… An)t = Ant, ……., A2t, A1t (reversal law for transpose)
Ques. What are the Properties Of Symmetric & Skew Matrix? (2 marks)
Ans. Property 1: A is symmetric if At = As skew-symmetric if At = − A
Property 2: A + At is a symmetric matrix A − At is a skew symmetric matrix. Consider (A + At)t = At + (At)t = At + A = A + At
A + At is symmetric.
Similarly, we can prove that A − At is skew-symmetric.
Property 3: The sum of two symmetric matrices is a symmetric matrix and the sum of two skew-symmetric matrices is a skew-symmetric matrix. Let At = A; Bt = B where A & B have the same order. (A + B)t = A + B Similarly we can prove the other.
Property 4: If A & B are symmetric matrices then,
(a) AB + BA is a symmetric matrix
(b) AB − BA is a skew-symmetric matrix.
Property 5: Every square matrix can be uniquely expressed as a sum of a symmetric and a skew-symmetric matrix.
Ques. What are the properties of Addition of Matrices? (2 marks)
Ans. If A, B and C are three matrices of order m x n, then
- Commutative Law A + B = B + A
- Associative Law (A + B) + C = A + (B + C)
- Existence of Additive Identity A zero matrix (0) of order m x n (same as of A), is the additive identity, if
A + 0 = A = 0 + A - Existence of Additive Inverse If A is a square matrix, then the matrix (- A) is called additive inverse, if
A + ( – A) = 0 = (- A) + A - Cancellation Law
A + B = A + C ⇒ B = C (left cancellation law)
B + A = C + A ⇒ B = C (right cancellation law)
Ques. What is an Orthogonal Matrix? (2 marks)
Ans. A square matrix is said to be orthogonal when it comprises real elements and its transpose is equal to its inverse. In other words, when the product of the real square matrix and its transpose is equal to an identity matrix, the real square matrix is said to be an orthogonal matrix.
Let A be the square matrix, AT is the transpose of A and A-1 is the inverse of A.
If
AT = A-1
then
AAT = ATA = I
Here, I is the identity matrix.
Ques. What is the inverse of a Matrix? (2 marks)
Ans. The product of a matrix and its inverse gives an identity matrix. The inverse of matrix A is denoted by A-1. The inverse of a matrix exists only for square matrices with non-zero determinant values.
A-1 = adj A / |A|, where |A| ≠ 0
Ques. What is the Dot Product of Orthogonal Matrices? (2 marks)
Ans. When two vectors are perpendicular to each other, their dot product is zero. So, the dot product of two orthogonal matrices is zero, as their vectors are perpendicular to each other.
Ques. What is a Symmetric Matrix? (2 marks)
Ans. A symmetric matrix is a square matrix whose transpose is equal to the matrix itself.
A = AT
Here, if you check the AT, it will be the same as A. So, A is a symmetric matrix.
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