diff --git a/src/function/arithmetic/multiply.js b/src/function/arithmetic/multiply.js
index 4ea04a1da7..801b9e23f3 100644
--- a/src/function/arithmetic/multiply.js
+++ b/src/function/arithmetic/multiply.js
@@ -11,10 +11,11 @@ const dependencies = [
   'matrix',
   'addScalar',
   'multiplyScalar',
-  'equalScalar'
+  'equalScalar',
+  'dot'
 ]
 
-export const createMultiply = /* #__PURE__ */ factory(name, dependencies, ({ typed, matrix, addScalar, multiplyScalar, equalScalar }) => {
+export const createMultiply = /* #__PURE__ */ factory(name, dependencies, ({ typed, matrix, addScalar, multiplyScalar, equalScalar, dot }) => {
   const algorithm11 = createAlgorithm11({ typed, equalScalar })
   const algorithm14 = createAlgorithm14({ typed })
 
@@ -201,38 +202,7 @@ export const createMultiply = /* #__PURE__ */ factory(name, dependencies, ({ typ
   function _multiplyVectorVector (a, b, n) {
     // check empty vector
     if (n === 0) { throw new Error('Cannot multiply two empty vectors') }
-
-    // a dense
-    const adata = a._data
-    const adt = a._datatype
-    // b dense
-    const bdata = b._data
-    const bdt = b._datatype
-
-    // datatype
-    let dt
-    // addScalar signature to use
-    let af = addScalar
-    // multiplyScalar signature to use
-    let mf = multiplyScalar
-
-    // process data types
-    if (adt && bdt && adt === bdt && typeof adt === 'string') {
-      // datatype
-      dt = adt
-      // find signatures that matches (dt, dt)
-      af = typed.find(addScalar, [dt, dt])
-      mf = typed.find(multiplyScalar, [dt, dt])
-    }
-
-    // result (do not initialize it with zero)
-    let c = mf(adata[0], bdata[0])
-    // loop data
-    for (let i = 1; i < n; i++) {
-      // multiply and accumulate
-      c = af(c, mf(adata[i], bdata[i]))
-    }
-    return c
+    return dot(a, b)
   }
 
   /**
diff --git a/src/function/matrix/dot.js b/src/function/matrix/dot.js
index 7e7a7daf98..a03f39c7c8 100644
--- a/src/function/matrix/dot.js
+++ b/src/function/matrix/dot.js
@@ -1,15 +1,15 @@
-import { arraySize as size } from '../../utils/array'
 import { factory } from '../../utils/factory'
+import { isMatrix } from '../../utils/is'
 
 const name = 'dot'
-const dependencies = ['typed', 'add', 'multiply']
+const dependencies = ['typed', 'addScalar', 'multiplyScalar', 'conj', 'size']
 
-export const createDot = /* #__PURE__ */ factory(name, dependencies, ({ typed, add, multiply }) => {
+export const createDot = /* #__PURE__ */ factory(name, dependencies, ({ typed, addScalar, multiplyScalar, conj, size }) => {
   /**
    * Calculate the dot product of two vectors. The dot product of
-   * `A = [a1, a2, a3, ..., an]` and `B = [b1, b2, b3, ..., bn]` is defined as:
+   * `A = [a1, a2, ..., an]` and `B = [b1, b2, ..., bn]` is defined as:
    *
-   *    dot(A, B) = a1 * b1 + a2 * b2 + a3 * b3 + ... + an * bn
+   *    dot(A, B) = conj(a1) * b1 + conj(a2) * b2 + ... + conj(an) * bn
    *
    * Syntax:
    *
@@ -29,43 +29,138 @@ export const createDot = /* #__PURE__ */ factory(name, dependencies, ({ typed, a
    * @return {number}               Returns the dot product of `x` and `y`
    */
   return typed(name, {
-    'Matrix, Matrix': function (x, y) {
-      return _dot(x.toArray(), y.toArray())
-    },
+    'Array | DenseMatrix, Array | DenseMatrix': _denseDot,
+    'SparseMatrix, SparseMatrix': _sparseDot
+  })
 
-    'Matrix, Array': function (x, y) {
-      return _dot(x.toArray(), y)
-    },
+  function _validateDim (x, y) {
+    const xSize = _size(x)
+    const ySize = _size(y)
+    let xLen, yLen
 
-    'Array, Matrix': function (x, y) {
-      return _dot(x, y.toArray())
-    },
+    if (xSize.length === 1) {
+      xLen = xSize[0]
+    } else if (xSize.length === 2 && xSize[1] === 1) {
+      xLen = xSize[0]
+    } else {
+      throw new RangeError('Expected a column vector, instead got a matrix of size (' + xSize.join(', ') + ')')
+    }
 
-    'Array, Array': _dot
-  })
+    if (ySize.length === 1) {
+      yLen = ySize[0]
+    } else if (ySize.length === 2 && ySize[1] === 1) {
+      yLen = ySize[0]
+    } else {
+      throw new RangeError('Expected a column vector, instead got a matrix of size (' + ySize.join(', ') + ')')
+    }
 
-  /**
-   * Calculate the dot product for two arrays
-   * @param {Array} x  First vector
-   * @param {Array} y  Second vector
-   * @returns {number} Returns the dot product of x and y
-   * @private
-   */
-  // TODO: double code with math.multiply
-  function _dot (x, y) {
-    const xSize = size(x)
-    const ySize = size(y)
-    const len = xSize[0]
-
-    if (xSize.length !== 1 || ySize.length !== 1) throw new RangeError('Vector expected') // TODO: better error message
-    if (xSize[0] !== ySize[0]) throw new RangeError('Vectors must have equal length (' + xSize[0] + ' != ' + ySize[0] + ')')
-    if (len === 0) throw new RangeError('Cannot calculate the dot product of empty vectors')
-
-    let prod = 0
-    for (let i = 0; i < len; i++) {
-      prod = add(prod, multiply(x[i], y[i]))
+    if (xLen !== yLen) throw new RangeError('Vectors must have equal length (' + xLen + ' != ' + yLen + ')')
+    if (xLen === 0) throw new RangeError('Cannot calculate the dot product of empty vectors')
+
+    return xLen
+  }
+
+  function _denseDot (a, b) {
+    const N = _validateDim(a, b)
+
+    const adata = isMatrix(a) ? a._data : a
+    const adt = isMatrix(a) ? a._datatype : undefined
+
+    const bdata = isMatrix(b) ? b._data : b
+    const bdt = isMatrix(b) ? b._datatype : undefined
+
+    // are these 2-dimensional column vectors? (as opposed to 1-dimensional vectors)
+    const aIsColumn = _size(a).length === 2
+    const bIsColumn = _size(b).length === 2
+
+    let add = addScalar
+    let mul = multiplyScalar
+
+    // process data types
+    if (adt && bdt && adt === bdt && typeof adt === 'string') {
+      const dt = adt
+      // find signatures that matches (dt, dt)
+      add = typed.find(addScalar, [dt, dt])
+      mul = typed.find(multiplyScalar, [dt, dt])
+    }
+
+    // both vectors 1-dimensional
+    if (!aIsColumn && !bIsColumn) {
+      let c = mul(conj(adata[0]), bdata[0])
+      for (let i = 1; i < N; i++) {
+        c = add(c, mul(conj(adata[i]), bdata[i]))
+      }
+      return c
     }
 
-    return prod
+    // a is 1-dim, b is column
+    if (!aIsColumn && bIsColumn) {
+      let c = mul(conj(adata[0]), bdata[0][0])
+      for (let i = 1; i < N; i++) {
+        c = add(c, mul(conj(adata[i]), bdata[i][0]))
+      }
+      return c
+    }
+
+    // a is column, b is 1-dim
+    if (aIsColumn && !bIsColumn) {
+      let c = mul(conj(adata[0][0]), bdata[0])
+      for (let i = 1; i < N; i++) {
+        c = add(c, mul(conj(adata[i][0]), bdata[i]))
+      }
+      return c
+    }
+
+    // both vectors are column
+    if (aIsColumn && bIsColumn) {
+      let c = mul(conj(adata[0][0]), bdata[0][0])
+      for (let i = 1; i < N; i++) {
+        c = add(c, mul(conj(adata[i][0]), bdata[i][0]))
+      }
+      return c
+    }
+  }
+
+  function _sparseDot (x, y) {
+    _validateDim(x, y)
+
+    const xindex = x._index
+    const xvalues = x._values
+
+    const yindex = y._index
+    const yvalues = y._values
+
+    // TODO optimize add & mul using datatype
+    let c = 0
+    const add = addScalar
+    const mul = multiplyScalar
+
+    let i = 0
+    let j = 0
+    while (i < xindex.length && j < yindex.length) {
+      const I = xindex[i]
+      const J = yindex[j]
+
+      if (I < J) {
+        i++
+        continue
+      }
+      if (I > J) {
+        j++
+        continue
+      }
+      if (I === J) {
+        c = add(c, mul(xvalues[i], yvalues[j]))
+        i++
+        j++
+      }
+    }
+
+    return c
+  }
+
+  // TODO remove this once #1771 is fixed
+  function _size (x) {
+    return isMatrix(x) ? x.size() : size(x)
   }
 })
diff --git a/test/unit-tests/function/arithmetic/multiply.test.js b/test/unit-tests/function/arithmetic/multiply.test.js
index 7a12087c44..3c3d3b699e 100644
--- a/test/unit-tests/function/arithmetic/multiply.test.js
+++ b/test/unit-tests/function/arithmetic/multiply.test.js
@@ -285,6 +285,14 @@ describe('multiply', function () {
       approx.deepEqual(multiply(matrix(a), matrix(b)), 32)
     })
 
+    it('should conjugate the first argument in dot product', function () {
+      const a = [complex(1, 2), complex(3, 4)]
+      const b = [complex(5, 6), complex(7, 8)]
+
+      approx.deepEqual(multiply(a, b), complex(70, -8))
+      approx.deepEqual(multiply(matrix(a), matrix(b)), complex(70, -8))
+    })
+
     it('should multiply row vector x column vector', function () {
       const v = [[1, 2, 3, 0, 0, 5, 6]]
 
diff --git a/test/unit-tests/function/matrix/dot.test.js b/test/unit-tests/function/matrix/dot.test.js
index e2c7d6704b..a3de75795c 100644
--- a/test/unit-tests/function/matrix/dot.test.js
+++ b/test/unit-tests/function/matrix/dot.test.js
@@ -1,40 +1,93 @@
 import assert from 'assert'
 import math from '../../../../src/bundleAny'
 
+const dot = math.dot
+const matrix = math.matrix
+const sparse = math.sparse
+const complex = math.complex
+
 describe('dot', function () {
-  it('should calculate dot product for two arrays', function () {
-    assert.strictEqual(math.dot([2, 4, 1], [2, 2, 3]), 15)
-    assert.strictEqual(math.dot([7, 3], [2, 4]), 26)
+  it('should calculate dot product for two 1-dim arrays', function () {
+    assert.strictEqual(dot([2, 4, 1], [2, 2, 3]), 15)
+    assert.strictEqual(dot([7, 3], [2, 4]), 26)
+  })
+
+  it('should calculate dot product for two column arrays', function () {
+    assert.strictEqual(dot([[2], [4], [1]], [[2], [2], [3]]), 15)
+    assert.strictEqual(dot([[7], [3]], [[2], [4]]), 26)
+  })
+
+  it('should calculate dot product for two 1-dim vectors', function () {
+    assert.strictEqual(dot(matrix([2, 4, 1]), matrix([2, 2, 3])), 15)
+    assert.strictEqual(dot(matrix([7, 3]), matrix([2, 4])), 26)
+  })
+
+  it('should calculate dot product for two column vectors', function () {
+    assert.strictEqual(dot(matrix([[2], [4], [1]]), matrix([[2], [2], [3]])), 15)
+    assert.strictEqual(dot(matrix([[7], [3]]), matrix([[2], [4]])), 26)
+  })
+
+  it('should calculate dot product for mixed 1-dim arrays and column arrays', function () {
+    assert.strictEqual(dot([2, 4, 1], [[2], [2], [3]]), 15)
+    assert.strictEqual(dot([[7], [3]], [2, 4]), 26)
+  })
+
+  it('should calculate dot product for mixed 1-dim arrays and 1-dim vectors', function () {
+    assert.strictEqual(dot([2, 4, 1], matrix([2, 2, 3])), 15)
+    assert.strictEqual(dot(matrix([7, 3]), [2, 4]), 26)
+  })
+
+  it('should calculate dot product for mixed 1-dim arrays and column vectors', function () {
+    assert.strictEqual(dot([2, 4, 1], matrix([[2], [2], [3]])), 15)
+    assert.strictEqual(dot(matrix([[7], [3]]), [2, 4]), 26)
+  })
+
+  it('should calculate dot product for mixed column arrays and 1-dim vectors', function () {
+    assert.strictEqual(dot([[2], [4], [1]], matrix([2, 2, 3])), 15)
+    assert.strictEqual(dot(matrix([7, 3]), [[2], [4]]), 26)
   })
 
-  it('should calculate dot product for two matrices', function () {
-    assert.strictEqual(math.dot(math.matrix([2, 4, 1]), math.matrix([2, 2, 3])), 15)
-    assert.strictEqual(math.dot(math.matrix([7, 3]), math.matrix([2, 4])), 26)
+  it('should calculate dot product for mixed column arrays and column vectors', function () {
+    assert.strictEqual(dot([[2], [4], [1]], matrix([[2], [2], [3]])), 15)
+    assert.strictEqual(dot(matrix([[7], [3]]), [[2], [4]]), 26)
   })
 
-  it('should calculate dot product for mixed arrays and matrices', function () {
-    assert.strictEqual(math.dot([2, 4, 1], math.matrix([2, 2, 3])), 15)
-    assert.strictEqual(math.dot(math.matrix([7, 3]), [2, 4]), 26)
+  it('should calculate dot product for mixed 1-dim vectors and column vectors', function () {
+    assert.strictEqual(dot(matrix([2, 4, 1]), matrix([[2], [2], [3]])), 15)
+    assert.strictEqual(dot(matrix([[7], [3]]), matrix([2, 4])), 26)
+  })
+
+  it('should calculate dot product for sparse vectors', function () {
+    assert.strictEqual(dot(sparse([0, 0, 2, 4, 4, 1]), sparse([1, 0, 2, 2, 0, 3])), 15)
+    assert.strictEqual(dot(sparse([7, 1, 2, 3]), sparse([2, 0, 0, 4])), 26)
   })
 
   it('should throw an error for unsupported types of arguments', function () {
-    assert.throws(function () { math.dot([2, 4, 1], 2) }, TypeError)
+    assert.throws(function () { dot([2, 4, 1], 2) }, TypeError)
   })
 
   it('should throw an error for multi dimensional matrix input', function () {
-    assert.throws(function () { math.dot([[1, 2], [3, 4]], [[1, 2], [3, 4]]) }, /Vector expected/)
+    assert.throws(function () { dot([[1, 2], [3, 4]], [[1, 2], [3, 4]]) }, /Expected a column vector, instead got a matrix of size \(2, 2\)/)
   })
 
   it('should throw an error in case of vectors with unequal length', function () {
-    assert.throws(function () { math.dot([2, 3], [1, 2, 3]) }, /Vectors must have equal length \(2 != 3\)/)
+    assert.throws(function () { dot([2, 3], [1, 2, 3]) }, /Vectors must have equal length \(2 != 3\)/)
   })
 
   it('should throw an error in case of empty vectors', function () {
-    assert.throws(function () { math.dot([], []) }, /Cannot calculate the dot product of empty vectors/)
+    assert.throws(function () { dot([], []) }, /Cannot calculate the dot product of empty vectors/)
   })
 
   it('should LaTeX dot', function () {
     const expression = math.parse('dot([1,2],[3,4])')
     assert.strictEqual(expression.toTex(), '\\left(\\begin{bmatrix}1\\\\2\\\\\\end{bmatrix}\\cdot\\begin{bmatrix}3\\\\4\\\\\\end{bmatrix}\\right)')
   })
+
+  it('should be antilinear in the first argument', function () {
+    const I = complex(0, 1)
+    assert.deepStrictEqual(dot([I, 2], [1, I]), I)
+
+    const v = matrix([2, I, 1])
+    assert.deepStrictEqual(dot(v, v).sqrt(), complex(math.norm(v)))
+  })
 })